Systems of Equations and Inequalities: Cheat Sheet

Essential Concepts

Systems of Linear Equations: Two Variables

  • A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
  • One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes.
  • Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation.
  • A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables.
  • It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together.
  • Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect.
  • The solution to a system of dependent equations will always be true because both equations describe the same line.
  • Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit.

Systems of Linear Equations: Three Variables

  • A solution set is an ordered triple [latex]\left\{\left(x,y,z\right)\right\}[/latex] that represents the intersection of three planes in space.
  • A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation.
  • Systems of three equations in three variables are useful for solving many different types of real-world problems.
  • A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction.
  • Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
  • A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity.
  • Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

Systems of Nonlinear Equations and Inequalities

  • There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points.
  • There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the parabola; (3) two solutions, the line intersects the circle in two points.
  • There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:
    (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points.
  • An inequality is graphed in much the same way as an equation, except for [latex]>[/latex] or [latex]<[/latex], we draw a dashed line and shade the region containing the solution set.
  • Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities.

Partial Fraction Decomposition

  • Decompose [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] by writing the partial fractions as [latex]\frac{A}{{a}_{1}x+{b}_{1}}+\frac{B}{{a}_{2}x+{b}_{2}}[/latex]. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations.
  • The decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] with repeated linear factors must account for the factors of the denominator in increasing powers.
  • The decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in [latex]\frac{A}{x}+\frac{Bx+C}{\left(a{x}^{2}+bx+c\right)}[/latex].
  • In the decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex], where [latex]Q\left(x\right)[/latex] has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

    [latex]\frac{Ax+B}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\cdots \text{+}\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}[/latex].

Glossary

addition method
An algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable
break-even point
The point at which a cost function intersects a revenue function; where profit is zero
consistent system
A system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system
cost function
The function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs
dependent system
A system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system
inconsistent system
A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common
independent system
A system of linear equations with exactly one solution pair [latex]\left(x,y\right)[/latex]
partial fractions
 the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression
partial fraction decomposition
the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions
profit function
The profit function is written as [latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex], revenue minus cost
revenue function
The function that is used to calculate revenue, simply written as [latex]R=xp[/latex], where [latex]x=[/latex] quantity and [latex]p=[/latex] price
solution set
the set of all ordered pairs or triples that satisfy all equations in a system of equations
substitution method
An algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable
system of linear equations
A set of two or more equations in two or more variables that must be considered simultaneously