Algebra · Quick Reference

Systems of Equations Cheat Sheet

Algebra I · Lesson 5/17

In one line: a system of equations is a set of two or more equations sharing the same variables — solving it means finding the values that satisfy every equation at once.

Key Ideas

1What Is a System. Two or more equations considered together; a solution must make every equation true simultaneously.

2Solving by Substitution. Solve one equation for one variable, then substitute that expression into the other equation to solve for the remaining variable.

3Solving by Elimination. Add or subtract the equations (after scaling if needed) to cancel out one variable, leaving an equation in a single variable.

4Types of Solutions. A system can have exactly one solution (lines cross once), no solution (parallel lines, never cross), or infinitely many solutions (same line, written differently).

5Graphical Interpretation. The solution to a system of two linear equations is the point where their two graphed lines intersect.

Worked Examples

Solve the system: x + y = 10, x - y = 2, using elimination.

x = 6, y = 4

Solve the system: y = 2x + 1, y = x + 4, using substitution.

x = 3, y = 7

Solve the system: 2x + y = 8, x - y = 1, using elimination.

x = 3, y = 2

Practice Yourself

Solve: x+y=12, x-y=4.

x=8, y=4

Solve: y=3x, y=x+8.

x=4, y=12

Solve: 2x+y=10, x-y=2.

x=4, y=2

What does it mean if a system has 'no solution'?

The lines are parallel and never intersect