Essential Concepts

Systems of Linear Equations in Two Variables

System of Linear Equations

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.

Solving by Graphing

1. Graph both equations on the same coordinate plane
2. Identify the point of intersection (if it exists)
3. Check the solution in both original equations

Solving by Substitution

1. Solve one equation for one variable
2. Substitute the expression into the other equation
3. Solve for the remaining variable
4. Back-substitute to find the other variable
5. Check the solution in both equations

Solving by Elimination (Addition Method)

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

Types of Systems

Systems of Linear Equations in Three Variables

Solution Process

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

Types of Solutions for Three-Variable Systems

Expressing Solutions of Dependent Systems

For dependent systems, express the solution in terms of one variable:

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

Systems of Nonlinear Equations and Inequalities

System of Nonlinear Equations

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

Possible Solutions: Line and Parabola

Possible Solutions: Line and Circle

Solving by Substitution

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

Graphing Nonlinear Inequalities

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

Graphing Systems of Nonlinear Inequalities

1. Find intersection points by solving the corresponding system of equations
2. Graph each inequality separately
3. Identify the region where all shaded regions overlap

Partial Fractions

Decomposition with Nonrepeated Linear Factors

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

[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]

Strategy:

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

Decomposition with Repeated Linear Factors

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

[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]

Decomposition with Nonrepeated Irreducible Quadratic Factors

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

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

Decomposition with Repeated Irreducible Quadratic Factors

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

[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]

General Strategy for Partial Fractions

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

Key Equations

Standard form of linear equation	[latex]Ax + By = C[/latex]
Slope-intercept form	[latex]y = mx + b[/latex]
Circle equation	[latex]x^2 + y^2 = r^2[/latex] or [latex](x - h)^2 + (y - k)^2 = r^2[/latex]
Parabola (vertical)	[latex]y = ax^2 + bx + c[/latex]

Glossary

addition method (elimination method)

An algebraic technique where equations are added (after multiplying by constants if needed) to eliminate one variable.

back-substitution

The process of substituting known variable values into previous equations to find remaining unknown variables.

break-even point

The point where a cost function intersects a revenue function; where profit is zero.

consistent system

A system that has at least one solution (either unique or infinitely many); can be independent or dependent.

cost function

A function describing the costs of doing business; typically has fixed costs and variable costs.

dependent system

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.

extraneous solution

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

inconsistent system

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

independent system

A system with exactly one solution; graphically, the lines or planes intersect at exactly one point.

Gaussian elimination

A systematic method for solving systems of linear equations using row operations to achieve upper triangular form.

irreducible quadratic factor

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

nonlinear inequality

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

ordered triple

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

partial fraction decomposition

The process of breaking down a simplified rational expression into a sum or difference of simpler rational expressions.

partial fractions

The individual simpler fractions that make up the sum or difference of a rational expression before combining them.

profit function

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

repeated factor

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

revenue function

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

solution set

The set of all ordered pairs or triples that satisfy all equations in a system.

substitution method

An algebraic technique where one equation is solved for one variable, and that expression is substituted into the other equation(s).

system of linear equations

A set of two or more linear equations with two or more variables that must be considered simultaneously.

system of nonlinear equations

A system containing at least one equation of degree larger than one (not linear).

system of nonlinear inequalities

A system of two or more inequalities in two or more variables with at least one nonlinear inequality.