{"id":694,"date":"2025-07-14T21:08:27","date_gmt":"2025-07-14T21:08:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=694"},"modified":"2026-01-15T17:06:41","modified_gmt":"2026-01-15T17:06:41","slug":"module-9-systems-of-equations-and-inequalities-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-9-systems-of-equations-and-inequalities-cheat-sheet\/","title":{"raw":"Systems of Equations and Inequalities: Cheat Sheet","rendered":"Systems of Equations and Inequalities: Cheat Sheet"},"content":{"raw":"

Essential Concepts<\/h2>\r\n

Systems of Linear Equations in Two Variables<\/h3>\r\n

System of Linear Equations<\/strong><\/p>\r\n

A system of linear equations consists of two or more linear equations with two or more variables considered simultaneously. The solution is an ordered pair [latex](x, y)[\/latex] that satisfies all equations in the system.<\/p>\r\n

Solving by Graphing<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. Graph both equations on the same coordinate plane<\/li>\r\n \t
  2. Identify the point of intersection (if it exists)<\/li>\r\n \t
  3. Check the solution in both original equations<\/li>\r\n<\/ol>\r\n

    Solving by Substitution<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. Solve one equation for one variable<\/li>\r\n \t
    2. Substitute the expression into the other equation<\/li>\r\n \t
    3. Solve for the remaining variable<\/li>\r\n \t
    4. Back-substitute to find the other variable<\/li>\r\n \t
    5. Check the solution in both equations<\/li>\r\n<\/ol>\r\n

      Solving by Elimination (Addition Method)<\/strong><\/p>\r\n\r\n

        \r\n \t
      1. Write both equations in standard form [latex]Ax + By = C[\/latex]<\/li>\r\n \t
      2. Multiply one or both equations by constants so that coefficients of one variable are opposites<\/li>\r\n \t
      3. Add the equations to eliminate one variable<\/li>\r\n \t
      4. Solve for the remaining variable<\/li>\r\n \t
      5. Substitute back to find the other variable<\/li>\r\n \t
      6. Check the solution<\/li>\r\n<\/ol>\r\n

        Types of Systems<\/strong><\/p>\r\n\r\n

        \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Type<\/th>\r\nNumber of Solutions<\/th>\r\nGraph Characteristics<\/th>\r\nRecognizing the System<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
        Independent<\/strong><\/td>\r\nExactly one solution [latex](x, y)[\/latex]<\/td>\r\nLines intersect at one point<\/td>\r\n<\/td>\r\n<\/tr>\r\n
        Inconsistent<\/strong><\/td>\r\nNo solution<\/td>\r\nLines are parallel (same slope, different y-intercepts)<\/td>\r\nDuring algebraic solution, you'll encounter a false statement such as [latex]0 = 5[\/latex] or [latex]3 = 7[\/latex].<\/td>\r\n<\/tr>\r\n
        Dependent<\/strong><\/td>\r\nInfinitely many solutions<\/td>\r\nLines are coincident (same line)<\/td>\r\nDuring algebraic solution, you'll encounter an identity such as [latex]0 = 0[\/latex] or [latex]5 = 5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

        Systems of Linear Equations in Three Variables<\/h3>\r\n

        Solution Process<\/strong><\/p>\r\n\r\n

          \r\n \t
        1. Eliminate one variable<\/strong> from two different pairs of equations to create two new equations with two variables<\/li>\r\n \t
        2. Solve the two-by-two system<\/strong> resulting from step 1<\/li>\r\n \t
        3. Back-substitute<\/strong> the known values into one of the original equations to find the third variable<\/li>\r\n \t
        4. Check<\/strong> the solution [latex](x, y, z)[\/latex] in all three original equations<\/li>\r\n<\/ol>\r\n

          Types of Solutions for Three-Variable Systems<\/strong><\/p>\r\n\r\n

          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Type<\/th>\r\nResult During Solution<\/th>\r\nGeometric Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          One solution<\/strong><\/td>\r\nUnique ordered triple [latex](x, y, z)[\/latex]<\/td>\r\nThree planes intersect at one point<\/td>\r\n<\/tr>\r\n
          No solution (inconsistent)<\/strong><\/td>\r\nContradiction like [latex]0 = 3[\/latex]<\/td>\r\nThree parallel planes; two parallel planes and one intersecting; or planes that don't meet at a common point<\/td>\r\n<\/tr>\r\n
          Infinitely many solutions (dependent)<\/strong><\/td>\r\nIdentity like [latex]0 = 0[\/latex]<\/td>\r\nThree identical planes; three planes intersecting along a line; or two identical planes intersecting a third<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

          Expressing Solutions of Dependent Systems<\/strong><\/p>\r\n

          For dependent systems, express the solution in terms of one variable:<\/p>\r\n\r\n

            \r\n \t
          • [latex](x, y, z) = (x, mx + a, nx + b)[\/latex] if expressing in terms of [latex]x[\/latex]<\/li>\r\n \t
          • [latex](x, y, z) = (mz + a, nz + b, z)[\/latex] if expressing in terms of [latex]z[\/latex]<\/li>\r\n<\/ul>\r\n

            Systems of Nonlinear Equations and Inequalities<\/h3>\r\n

            System of Nonlinear Equations<\/strong><\/p>\r\n

            A system containing at least one equation of degree larger than one (not linear). Common types include systems with parabolas, circles, ellipses, and hyperbolas.<\/p>\r\n

            Possible Solutions: Line and Parabola<\/strong><\/p>\r\n\r\n

            \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            Number of Solutions<\/th>\r\nDescription<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
            No solution<\/td>\r\nLine doesn't intersect the parabola<\/td>\r\n<\/tr>\r\n
            One solution<\/td>\r\nLine is tangent to the parabola<\/td>\r\n<\/tr>\r\n
            Two solutions<\/td>\r\nLine crosses through the parabola<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

            Possible Solutions: Line and Circle<\/strong><\/p>\r\n\r\n

            \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            Number of Solutions<\/th>\r\nDescription<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
            No solution<\/td>\r\nLine doesn't intersect the circle<\/td>\r\n<\/tr>\r\n
            One solution<\/td>\r\nLine is tangent to the circle<\/td>\r\n<\/tr>\r\n
            Two solutions<\/td>\r\nLine crosses through the circle<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

            Solving by Substitution<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. Solve the linear equation (if present) for one variable<\/li>\r\n \t
            2. Substitute into the nonlinear equation<\/li>\r\n \t
            3. Solve the resulting equation (may be quadratic)<\/li>\r\n \t
            4. Back-substitute to find the other variable<\/li>\r\n \t
            5. Check for extraneous solutions<\/li>\r\n<\/ol>\r\n

              Graphing Nonlinear Inequalities<\/strong><\/p>\r\n\r\n

                \r\n \t
              1. Convert inequality to equation by replacing inequality sign with equal sign<\/li>\r\n \t
              2. Graph the equation as a boundary:\r\n
                  \r\n \t
                • Use solid line\/curve for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]<\/li>\r\n \t
                • Use dashed line\/curve for [latex]<[\/latex] or [latex]>[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                • Choose a test point not on the boundary<\/li>\r\n \t
                • Shade the region where the inequality is true<\/li>\r\n<\/ol>\r\n

                  Graphing Systems of Nonlinear Inequalities<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  1. Find intersection points by solving the corresponding system of equations<\/li>\r\n \t
                  2. Graph each inequality separately<\/li>\r\n \t
                  3. Identify the region where all shaded regions overlap<\/li>\r\n<\/ol>\r\n

                    Partial Fractions<\/h3>\r\n

                    Decomposition with Nonrepeated Linear Factors<\/strong><\/p>\r\n

                    When [latex]Q(x)[\/latex] has distinct linear factors [latex](a_1x + b_1)(a_2x + b_2)\\cdots(a_nx + b_n)[\/latex]:<\/p>\r\n

                    [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{(a_1x + b_1)} + \\frac{A_2}{(a_2x + b_2)} + \\cdots + \\frac{A_n}{(a_nx + b_n)}[\/latex]<\/p>\r\n

                    Strategy:<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    1. Set up the decomposition with constant numerators [latex]A, B, C,[\/latex] etc.<\/li>\r\n \t
                    2. Multiply both sides by the common denominator<\/li>\r\n \t
                    3. Expand and collect like terms<\/li>\r\n \t
                    4. Set coefficients equal to create a system of equations<\/li>\r\n \t
                    5. Solve for the constants<\/li>\r\n<\/ol>\r\n

                      Decomposition with Repeated Linear Factors<\/strong><\/p>\r\n

                      When [latex]Q(x)[\/latex] has a repeated factor [latex](ax + b)^n[\/latex]:<\/p>\r\n

                      [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{(ax + b)} + \\frac{A_2}{(ax + b)^2} + \\frac{A_3}{(ax + b)^3} + \\cdots + \\frac{A_n}{(ax + b)^n}[\/latex]<\/p>\r\n

                      Decomposition with Nonrepeated Irreducible Quadratic Factors<\/strong><\/p>\r\n

                      When [latex]Q(x)[\/latex] has an irreducible quadratic factor [latex](ax^2 + bx + c)[\/latex] that cannot be factored:<\/p>\r\n

                      [latex]\\frac{P(x)}{Q(x)} = \\frac{A}{(dx + e)} + \\frac{Bx + C}{(ax^2 + bx + c)}[\/latex]<\/p>\r\n

                      Decomposition with Repeated Irreducible Quadratic Factors<\/strong><\/p>\r\n

                      When [latex]Q(x)[\/latex] has a repeated irreducible quadratic factor [latex](ax^2 + bx + c)^n[\/latex]:<\/p>\r\n

                      [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1x + B_1}{(ax^2 + bx + c)} + \\frac{A_2x + B_2}{(ax^2 + bx + c)^2} + \\cdots + \\frac{A_nx + B_n}{(ax^2 + bx + c)^n}[\/latex]<\/p>\r\n

                      General Strategy for Partial Fractions<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      1. Factor the denominator [latex]Q(x)[\/latex] completely<\/li>\r\n \t
                      2. Set up the partial fraction decomposition based on the types of factors:\r\n
                          \r\n \t
                        • Constant numerators for linear factors<\/li>\r\n \t
                        • Linear numerators for quadratic factors<\/li>\r\n \t
                        • Include all powers for repeated factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                        • Multiply both sides by [latex]Q(x)[\/latex] to clear denominators<\/li>\r\n \t
                        • Expand and collect like terms<\/li>\r\n \t
                        • Match coefficients to create a system of equations<\/li>\r\n \t
                        • Solve the system (substitution, elimination, or strategic x-values)<\/li>\r\n \t
                        • Write the final decomposition<\/li>\r\n<\/ol>\r\n

                          Key Equations<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                          Standard form of linear equation<\/strong><\/td>\r\n[latex]Ax + By = C[\/latex]<\/td>\r\n<\/tr>\r\n
                          Slope-intercept form<\/strong><\/td>\r\n[latex]y = mx + b[\/latex]<\/td>\r\n<\/tr>\r\n
                          Circle equation<\/strong><\/td>\r\n[latex]x^2 + y^2 = r^2[\/latex] or [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/td>\r\n<\/tr>\r\n
                          Parabola (vertical)<\/strong><\/td>\r\n[latex]y = ax^2 + bx + c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                          Glossary<\/h2>\r\n

                          addition method (elimination method)<\/strong><\/p>\r\n

                          An algebraic technique where equations are added (after multiplying by constants if needed) to eliminate one variable.<\/p>\r\n

                          back-substitution<\/strong><\/p>\r\n

                          The process of substituting known variable values into previous equations to find remaining unknown variables.<\/p>\r\n

                          break-even point<\/strong><\/p>\r\n

                          The point where a cost function intersects a revenue function; where profit is zero.<\/p>\r\n

                          consistent system<\/strong><\/p>\r\n

                          A system that has at least one solution (either unique or infinitely many); can be independent or dependent.<\/p>\r\n

                          cost function<\/strong><\/p>\r\n

                          A function describing the costs of doing business; typically has fixed costs and variable costs.<\/p>\r\n

                          dependent system<\/strong><\/p>\r\n

                          A system with infinitely many solutions; for two variables, the lines are coincident (the same line); for three variables, the planes intersect along a line or are identical.<\/p>\r\n

                          extraneous solution<\/strong><\/p>\r\n

                          A solution that emerges from the algebraic process but doesn't satisfy the original equation; common when squaring both sides or solving nonlinear systems.<\/p>\r\n

                          inconsistent system<\/strong><\/p>\r\n

                          A system with no solution; for two variables, the lines are parallel; for three variables, the planes don't meet at a common point.<\/p>\r\n

                          independent system<\/strong><\/p>\r\n

                          A system with exactly one solution; graphically, the lines or planes intersect at exactly one point.<\/p>\r\n

                          Gaussian elimination<\/strong><\/p>\r\n

                          A systematic method for solving systems of linear equations using row operations to achieve upper triangular form.<\/p>\r\n

                          irreducible quadratic factor<\/strong><\/p>\r\n

                          A quadratic expression that cannot be factored into linear factors with real coefficients (the discriminant [latex]b^2 - 4ac < 0[\/latex]).<\/p>\r\n

                          nonlinear inequality<\/strong><\/p>\r\n

                          An inequality containing a nonlinear expression (such as [latex]x^2[\/latex], [latex]xy[\/latex], etc.).<\/p>\r\n

                          ordered triple<\/strong><\/p>\r\n

                          A solution [latex](x, y, z)[\/latex] to a system of three equations in three variables.<\/p>\r\n

                          partial fraction decomposition<\/strong><\/p>\r\n

                          The process of breaking down a simplified rational expression into a sum or difference of simpler rational expressions.<\/p>\r\n

                          partial fractions<\/strong><\/p>\r\n

                          The individual simpler fractions that make up the sum or difference of a rational expression before combining them.<\/p>\r\n

                          profit function<\/strong><\/p>\r\n

                          The difference between revenue and cost: [latex]P(x) = R(x) - C(x)[\/latex].<\/p>\r\n

                          repeated factor<\/strong><\/p>\r\n

                          A factor in the denominator that appears more than once, written as [latex](ax + b)^n[\/latex] where [latex]n > 1[\/latex].<\/p>\r\n

                          revenue function<\/strong><\/p>\r\n

                          A function calculating revenue, written as [latex]R = xp[\/latex] where [latex]x[\/latex] is quantity and [latex]p[\/latex] is price.<\/p>\r\n

                          solution set<\/strong><\/p>\r\n

                          The set of all ordered pairs or triples that satisfy all equations in a system.<\/p>\r\n

                          substitution method<\/strong><\/p>\r\n

                          An algebraic technique where one equation is solved for one variable, and that expression is substituted into the other equation(s).<\/p>\r\n

                          system of linear equations<\/strong><\/p>\r\n

                          A set of two or more linear equations with two or more variables that must be considered simultaneously.<\/p>\r\n

                          system of nonlinear equations<\/strong><\/p>\r\n

                          A system containing at least one equation of degree larger than one (not linear).<\/p>\r\n

                          system of nonlinear inequalities<\/strong><\/p>\r\n

                          A system of two or more inequalities in two or more variables with at least one nonlinear inequality.<\/p>","rendered":"

                          Essential Concepts<\/h2>\n

                          Systems of Linear Equations in Two Variables<\/h3>\n

                          System of Linear Equations<\/strong><\/p>\n

                          A system of linear equations consists of two or more linear equations with two or more variables considered simultaneously. The solution is an ordered pair [latex](x, y)[\/latex] that satisfies all equations in the system.<\/p>\n

                          Solving by Graphing<\/strong><\/p>\n

                            \n
                          1. Graph both equations on the same coordinate plane<\/li>\n
                          2. Identify the point of intersection (if it exists)<\/li>\n
                          3. Check the solution in both original equations<\/li>\n<\/ol>\n

                            Solving by Substitution<\/strong><\/p>\n

                              \n
                            1. Solve one equation for one variable<\/li>\n
                            2. Substitute the expression into the other equation<\/li>\n
                            3. Solve for the remaining variable<\/li>\n
                            4. Back-substitute to find the other variable<\/li>\n
                            5. Check the solution in both equations<\/li>\n<\/ol>\n

                              Solving by Elimination (Addition Method)<\/strong><\/p>\n

                                \n
                              1. Write both equations in standard form [latex]Ax + By = C[\/latex]<\/li>\n
                              2. Multiply one or both equations by constants so that coefficients of one variable are opposites<\/li>\n
                              3. Add the equations to eliminate one variable<\/li>\n
                              4. Solve for the remaining variable<\/li>\n
                              5. Substitute back to find the other variable<\/li>\n
                              6. Check the solution<\/li>\n<\/ol>\n

                                Types of Systems<\/strong><\/p>\n

                                \n\n\n\n\n\n\n\n
                                Type<\/th>\nNumber of Solutions<\/th>\nGraph Characteristics<\/th>\nRecognizing the System<\/th>\n<\/tr>\n<\/thead>\n
                                Independent<\/strong><\/td>\nExactly one solution [latex](x, y)[\/latex]<\/td>\nLines intersect at one point<\/td>\n<\/td>\n<\/tr>\n
                                Inconsistent<\/strong><\/td>\nNo solution<\/td>\nLines are parallel (same slope, different y-intercepts)<\/td>\nDuring algebraic solution, you’ll encounter a false statement such as [latex]0 = 5[\/latex] or [latex]3 = 7[\/latex].<\/td>\n<\/tr>\n
                                Dependent<\/strong><\/td>\nInfinitely many solutions<\/td>\nLines are coincident (same line)<\/td>\nDuring algebraic solution, you’ll encounter an identity such as [latex]0 = 0[\/latex] or [latex]5 = 5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                Systems of Linear Equations in Three Variables<\/h3>\n

                                Solution Process<\/strong><\/p>\n

                                  \n
                                1. Eliminate one variable<\/strong> from two different pairs of equations to create two new equations with two variables<\/li>\n
                                2. Solve the two-by-two system<\/strong> resulting from step 1<\/li>\n
                                3. Back-substitute<\/strong> the known values into one of the original equations to find the third variable<\/li>\n
                                4. Check<\/strong> the solution [latex](x, y, z)[\/latex] in all three original equations<\/li>\n<\/ol>\n

                                  Types of Solutions for Three-Variable Systems<\/strong><\/p>\n

                                  \n\n\n\n\n\n\n\n
                                  Type<\/th>\nResult During Solution<\/th>\nGeometric Interpretation<\/th>\n<\/tr>\n<\/thead>\n
                                  One solution<\/strong><\/td>\nUnique ordered triple [latex](x, y, z)[\/latex]<\/td>\nThree planes intersect at one point<\/td>\n<\/tr>\n
                                  No solution (inconsistent)<\/strong><\/td>\nContradiction like [latex]0 = 3[\/latex]<\/td>\nThree parallel planes; two parallel planes and one intersecting; or planes that don’t meet at a common point<\/td>\n<\/tr>\n
                                  Infinitely many solutions (dependent)<\/strong><\/td>\nIdentity like [latex]0 = 0[\/latex]<\/td>\nThree identical planes; three planes intersecting along a line; or two identical planes intersecting a third<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                  Expressing Solutions of Dependent Systems<\/strong><\/p>\n

                                  For dependent systems, express the solution in terms of one variable:<\/p>\n

                                    \n
                                  • [latex](x, y, z) = (x, mx + a, nx + b)[\/latex] if expressing in terms of [latex]x[\/latex]<\/li>\n
                                  • [latex](x, y, z) = (mz + a, nz + b, z)[\/latex] if expressing in terms of [latex]z[\/latex]<\/li>\n<\/ul>\n

                                    Systems of Nonlinear Equations and Inequalities<\/h3>\n

                                    System of Nonlinear Equations<\/strong><\/p>\n

                                    A system containing at least one equation of degree larger than one (not linear). Common types include systems with parabolas, circles, ellipses, and hyperbolas.<\/p>\n

                                    Possible Solutions: Line and Parabola<\/strong><\/p>\n

                                    \n\n\n\n\n\n\n\n
                                    Number of Solutions<\/th>\nDescription<\/th>\n<\/tr>\n<\/thead>\n
                                    No solution<\/td>\nLine doesn’t intersect the parabola<\/td>\n<\/tr>\n
                                    One solution<\/td>\nLine is tangent to the parabola<\/td>\n<\/tr>\n
                                    Two solutions<\/td>\nLine crosses through the parabola<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                    Possible Solutions: Line and Circle<\/strong><\/p>\n

                                    \n\n\n\n\n\n\n\n
                                    Number of Solutions<\/th>\nDescription<\/th>\n<\/tr>\n<\/thead>\n
                                    No solution<\/td>\nLine doesn’t intersect the circle<\/td>\n<\/tr>\n
                                    One solution<\/td>\nLine is tangent to the circle<\/td>\n<\/tr>\n
                                    Two solutions<\/td>\nLine crosses through the circle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                    Solving by Substitution<\/strong><\/p>\n

                                      \n
                                    1. Solve the linear equation (if present) for one variable<\/li>\n
                                    2. Substitute into the nonlinear equation<\/li>\n
                                    3. Solve the resulting equation (may be quadratic)<\/li>\n
                                    4. Back-substitute to find the other variable<\/li>\n
                                    5. Check for extraneous solutions<\/li>\n<\/ol>\n

                                      Graphing Nonlinear Inequalities<\/strong><\/p>\n

                                        \n
                                      1. Convert inequality to equation by replacing inequality sign with equal sign<\/li>\n
                                      2. Graph the equation as a boundary:\n
                                          \n
                                        • Use solid line\/curve for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]<\/li>\n
                                        • Use dashed line\/curve for [latex]<[\/latex] or [latex]>[\/latex]<\/li>\n<\/ul>\n<\/li>\n
                                        • Choose a test point not on the boundary<\/li>\n
                                        • Shade the region where the inequality is true<\/li>\n<\/ol>\n

                                          Graphing Systems of Nonlinear Inequalities<\/strong><\/p>\n

                                            \n
                                          1. Find intersection points by solving the corresponding system of equations<\/li>\n
                                          2. Graph each inequality separately<\/li>\n
                                          3. Identify the region where all shaded regions overlap<\/li>\n<\/ol>\n

                                            Partial Fractions<\/h3>\n

                                            Decomposition with Nonrepeated Linear Factors<\/strong><\/p>\n

                                            When [latex]Q(x)[\/latex] has distinct linear factors [latex](a_1x + b_1)(a_2x + b_2)\\cdots(a_nx + b_n)[\/latex]:<\/p>\n

                                            [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{(a_1x + b_1)} + \\frac{A_2}{(a_2x + b_2)} + \\cdots + \\frac{A_n}{(a_nx + b_n)}[\/latex]<\/p>\n

                                            Strategy:<\/strong><\/p>\n

                                              \n
                                            1. Set up the decomposition with constant numerators [latex]A, B, C,[\/latex] etc.<\/li>\n
                                            2. Multiply both sides by the common denominator<\/li>\n
                                            3. Expand and collect like terms<\/li>\n
                                            4. Set coefficients equal to create a system of equations<\/li>\n
                                            5. Solve for the constants<\/li>\n<\/ol>\n

                                              Decomposition with Repeated Linear Factors<\/strong><\/p>\n

                                              When [latex]Q(x)[\/latex] has a repeated factor [latex](ax + b)^n[\/latex]:<\/p>\n

                                              [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{(ax + b)} + \\frac{A_2}{(ax + b)^2} + \\frac{A_3}{(ax + b)^3} + \\cdots + \\frac{A_n}{(ax + b)^n}[\/latex]<\/p>\n

                                              Decomposition with Nonrepeated Irreducible Quadratic Factors<\/strong><\/p>\n

                                              When [latex]Q(x)[\/latex] has an irreducible quadratic factor [latex](ax^2 + bx + c)[\/latex] that cannot be factored:<\/p>\n

                                              [latex]\\frac{P(x)}{Q(x)} = \\frac{A}{(dx + e)} + \\frac{Bx + C}{(ax^2 + bx + c)}[\/latex]<\/p>\n

                                              Decomposition with Repeated Irreducible Quadratic Factors<\/strong><\/p>\n

                                              When [latex]Q(x)[\/latex] has a repeated irreducible quadratic factor [latex](ax^2 + bx + c)^n[\/latex]:<\/p>\n

                                              [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1x + B_1}{(ax^2 + bx + c)} + \\frac{A_2x + B_2}{(ax^2 + bx + c)^2} + \\cdots + \\frac{A_nx + B_n}{(ax^2 + bx + c)^n}[\/latex]<\/p>\n

                                              General Strategy for Partial Fractions<\/strong><\/p>\n

                                                \n
                                              1. Factor the denominator [latex]Q(x)[\/latex] completely<\/li>\n
                                              2. Set up the partial fraction decomposition based on the types of factors:\n
                                                  \n
                                                • Constant numerators for linear factors<\/li>\n
                                                • Linear numerators for quadratic factors<\/li>\n
                                                • Include all powers for repeated factors<\/li>\n<\/ul>\n<\/li>\n
                                                • Multiply both sides by [latex]Q(x)[\/latex] to clear denominators<\/li>\n
                                                • Expand and collect like terms<\/li>\n
                                                • Match coefficients to create a system of equations<\/li>\n
                                                • Solve the system (substitution, elimination, or strategic x-values)<\/li>\n
                                                • Write the final decomposition<\/li>\n<\/ol>\n

                                                  Key Equations<\/h2>\n\n\n\n\n\n\n
                                                  Standard form of linear equation<\/strong><\/td>\n[latex]Ax + By = C[\/latex]<\/td>\n<\/tr>\n
                                                  Slope-intercept form<\/strong><\/td>\n[latex]y = mx + b[\/latex]<\/td>\n<\/tr>\n
                                                  Circle equation<\/strong><\/td>\n[latex]x^2 + y^2 = r^2[\/latex] or [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/td>\n<\/tr>\n
                                                  Parabola (vertical)<\/strong><\/td>\n[latex]y = ax^2 + bx + c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                  Glossary<\/h2>\n

                                                  addition method (elimination method)<\/strong><\/p>\n

                                                  An algebraic technique where equations are added (after multiplying by constants if needed) to eliminate one variable.<\/p>\n

                                                  back-substitution<\/strong><\/p>\n

                                                  The process of substituting known variable values into previous equations to find remaining unknown variables.<\/p>\n

                                                  break-even point<\/strong><\/p>\n

                                                  The point where a cost function intersects a revenue function; where profit is zero.<\/p>\n

                                                  consistent system<\/strong><\/p>\n

                                                  A system that has at least one solution (either unique or infinitely many); can be independent or dependent.<\/p>\n

                                                  cost function<\/strong><\/p>\n

                                                  A function describing the costs of doing business; typically has fixed costs and variable costs.<\/p>\n

                                                  dependent system<\/strong><\/p>\n

                                                  A system with infinitely many solutions; for two variables, the lines are coincident (the same line); for three variables, the planes intersect along a line or are identical.<\/p>\n

                                                  extraneous solution<\/strong><\/p>\n

                                                  A solution that emerges from the algebraic process but doesn’t satisfy the original equation; common when squaring both sides or solving nonlinear systems.<\/p>\n

                                                  inconsistent system<\/strong><\/p>\n

                                                  A system with no solution; for two variables, the lines are parallel; for three variables, the planes don’t meet at a common point.<\/p>\n

                                                  independent system<\/strong><\/p>\n

                                                  A system with exactly one solution; graphically, the lines or planes intersect at exactly one point.<\/p>\n

                                                  Gaussian elimination<\/strong><\/p>\n

                                                  A systematic method for solving systems of linear equations using row operations to achieve upper triangular form.<\/p>\n

                                                  irreducible quadratic factor<\/strong><\/p>\n

                                                  A quadratic expression that cannot be factored into linear factors with real coefficients (the discriminant [latex]b^2 - 4ac < 0[\/latex]).<\/p>\n

                                                  nonlinear inequality<\/strong><\/p>\n

                                                  An inequality containing a nonlinear expression (such as [latex]x^2[\/latex], [latex]xy[\/latex], etc.).<\/p>\n

                                                  ordered triple<\/strong><\/p>\n

                                                  A solution [latex](x, y, z)[\/latex] to a system of three equations in three variables.<\/p>\n

                                                  partial fraction decomposition<\/strong><\/p>\n

                                                  The process of breaking down a simplified rational expression into a sum or difference of simpler rational expressions.<\/p>\n

                                                  partial fractions<\/strong><\/p>\n

                                                  The individual simpler fractions that make up the sum or difference of a rational expression before combining them.<\/p>\n

                                                  profit function<\/strong><\/p>\n

                                                  The difference between revenue and cost: [latex]P(x) = R(x) - C(x)[\/latex].<\/p>\n

                                                  repeated factor<\/strong><\/p>\n

                                                  A factor in the denominator that appears more than once, written as [latex](ax + b)^n[\/latex] where [latex]n > 1[\/latex].<\/p>\n

                                                  revenue function<\/strong><\/p>\n

                                                  A function calculating revenue, written as [latex]R = xp[\/latex] where [latex]x[\/latex] is quantity and [latex]p[\/latex] is price.<\/p>\n

                                                  solution set<\/strong><\/p>\n

                                                  The set of all ordered pairs or triples that satisfy all equations in a system.<\/p>\n

                                                  substitution method<\/strong><\/p>\n

                                                  An algebraic technique where one equation is solved for one variable, and that expression is substituted into the other equation(s).<\/p>\n

                                                  system of linear equations<\/strong><\/p>\n

                                                  A set of two or more linear equations with two or more variables that must be considered simultaneously.<\/p>\n

                                                  system of nonlinear equations<\/strong><\/p>\n

                                                  A system containing at least one equation of degree larger than one (not linear).<\/p>\n

                                                  system of nonlinear inequalities<\/strong><\/p>\n

                                                  A system of two or more inequalities in two or more variables with at least one nonlinear inequality.<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/694"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/694\/revisions"}],"predecessor-version":[{"id":5380,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/694\/revisions\/5380"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/694\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=694"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=694"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=694"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}