Solving Systems by Substitution Step by Step Cheat Sheet

Key Facts

• A solution to a system of equations is an ordered pair $(x, y)$ that makes both equations true.
• Substitution works best when one equation is already written as $x = \text{expression}$ or $y = \text{expression}$.
• If $y = 2x + 1$ and $3x + y = 16$, substitute to get $3x + (2x + 1) = 16$.
• After substitution, solve the resulting one-variable equation, such as $5x + 1 = 16$.
• Once one variable is found, substitute it into either original equation to find the other variable.
• Check the solution by substituting the ordered pair $(x, y)$ into both original equations.
• If substitution leads to a true statement like $4 = 4$, the system has infinitely many solutions.
• If substitution leads to a false statement like $7 = 2$, the system has no solution.

Vocabulary

System of equations

A system of equations is a set of two or more equations that use the same variables.

Substitution

Substitution is a solving method where one expression is replaced with an equal expression from another equation.

Ordered pair

An ordered pair $(x, y)$ gives the values of $x$ and $y$ that may solve a system.

Equivalent expression

Equivalent expressions have the same value for all allowed values of the variable.

No solution

A system has no solution when the equations represent lines that never intersect.

Infinitely many solutions

A system has infinitely many solutions when both equations represent the same line.