Solving Systems by Elimination Step by Step Cheat Sheet

Key Facts

• A system of two linear equations has a solution $(x, y)$ that makes both equations true.
• Elimination works best when equations are written in standard form $Ax + By = C$ with like terms aligned.
• To eliminate a variable, create opposite coefficients such as $5y$ and $-5y$, then add the equations.
• Multiplying an equation by a nonzero constant, such as $2(x + y) = 2(7)$, creates an equivalent equation with the same solutions.
• If the $x$-coefficients match exactly, subtract the equations to eliminate $x$ because $ax - ax = 0$.
• If the coefficients are not already matched, use the least common multiple, such as $\operatorname{LCM}(4, 6) = 12$, to choose helpful multipliers.
• After solving for one variable, substitute into an original equation, such as $x + 2(3) = 10$, to find the other variable.
• Always check the final ordered pair by substituting it into both original equations.

Vocabulary

System of equations

A set of two or more equations that are solved together using the same variables.

Elimination

A solving method that adds or subtracts equations to cancel one variable.

Coefficient

The number multiplying a variable, such as $-4$ in $-4x$.

Equivalent equation

An equation made by valid operations that has the same solutions as the original equation.

Ordered pair

A solution written as $(x, y)$ where the first number is the $x$-value and the second number is the $y$-value.

Check

The step of substituting the solution into the original equations to confirm both statements are true.