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College Algebra Unit 10: Systems of equations

A system is a set of equations solved together. The solution is the point (or points) that satisfies all of them at once.

Covers: Substitution/elimination, 3-variable systems, nonlinear systems, systems of inequalities

Study guide

A system of equations is solved when you find values that satisfy every equation at once. Substitution solves one equation for a variable and substitutes it into the others. Elimination adds or subtracts multiples of equations to cancel a variable. For two linear equations in two variables, the outcomes are one solution (intersecting lines), no solution (parallel lines; algebra gives a false statement like 0 = 5, an inconsistent system), or infinitely many (the same line; algebra gives 0 = 0, a dependent system).

Three-variable systems describe planes in space. Use elimination to remove the same variable from two different pairs of equations, leaving a two-variable system. Solve it, then back-substitute to find the third variable. Matrices and row reduction organize the same process for larger systems.

Nonlinear systems involve at least one nonlinear equation, like a line and a circle or two parabolas. Substitution usually works best, producing a quadratic or higher-degree equation. The number of real solutions equals the number of intersection points, which can be zero, one, two or more. Check each solution in both original equations.

Systems of inequalities are solved graphically. Graph each boundary (solid for ≤ or ≥, dashed for ), shade the side that satisfies each inequality, and the solution region is where all the shading overlaps. Linear programming uses the corners of such a region to find maximum or minimum values.

In a word problem, assign a variable to each unknown and write one equation for each fact given, such as the total amount invested and the total interest earned, then solve the system. To use matrices, write the coefficients and constants as an augmented matrix. Use row operations (swap rows, scale a row, add a multiple of one row to another) to reach row-echelon form, then back-substitute.

Key ideas

• Substitution: solve one equation for a variable and plug it into the other.
• Elimination: add or subtract equations to cancel a variable.
• A 2×2 linear system has one solution, no solution (parallel lines) or infinitely many (same line).
• Three-variable systems: eliminate one variable twice to get a 2×2 system.
• Systems of inequalities: graph each and shade the overlap.

Formulas and rules

Common mistakes

• Stopping after finding one variable.
• Calling 0 = 0 "no solution" when it means infinitely many.

Words to know

Elimination method

Add or subtract equations to cancel one variable.

Inconsistent system

Solving gives a false statement like 0 = 7: no solution.

Dependent system

Solving gives 0 = 0: infinitely many solutions.

No-solution system (graph)

Parallel lines.

Solution region (inequalities)

Where all shaded regions overlap.

Substitution method

Solve one equation for a variable and substitute it.

Three-variable system

Eliminate the same variable from two pairs, solve, back-substitute.

Nonlinear system

A system with a nonlinear equation, like a line and a circle.

Augmented matrix

A matrix of coefficients and constants representing a system.

Linear programming

Optimizing at the corner points of a feasible region.

Sample practice questions

25 more questions like these, from level 1 to 5, are in the practice set.