Lesson 1: Solving Systems of Linear Equations by Graphing

Property

When two or more linear equations are grouped together, they form a system of linear equations. Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair $(x, y)$. To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Examples

• Is $(2, 3)$ a solution to the system $\begin{cases} 3x - y = 3 \\ x + 2y = 8 \end{cases}$? For the first equation, $3(2) - 3 = 3$ is true. For the second, $2 + 2(3) = 8$ is true. Since it makes both true, $(2, 3)$ is a solution.
• Is $(-1, 5)$ a solution to the system $\begin{cases} 5x + y = 0 \\ 2x + y = 4 \end{cases}$? For the first equation, $5(-1) + 5 = 0$ is true. For the second, $2(-1) + 5 = 3 \neq 4$ is false. Therefore, $(-1, 5)$ is not a solution.
• Is $(4, -2)$ a solution to the system $\begin{cases} x + 3y = -2 \\ -2x - 5y = -2 \end{cases}$? For the first equation, $4 + 3(-2) = -2$ is true. For the second, $-2(4) - 5(-2) = -8 + 10 = 2 \neq -2$ is false. Therefore, $(4, -2)$ is not a solution.

Explanation

Think of a system's solution as a secret meeting point. It's the single ordered pair $(x, y)$ that exists on both lines at the same time. If a point only satisfies one equation, it hasn't arrived at the right spot.

Property

A solution to a system of equations is an ordered pair $(x, y)$ that satisfies each equation in the system simultaneously.

To check whether an ordered pair is a solution, substitute the coordinates into each equation to verify that they result in true statements.

Examples

• Verifying a Solution: Is $(3, 7)$ a solution to the system $y = 2x + 1$ and $y = 4x - 5$?

Check equation 1: $7 = 2(3) + 1$ becomes 7 = 7 (True).
Check equation 2: $7 = 4(3) - 5$ becomes 7 = 7 (True).
Yes, it is the solution.

• The "One Line" Trap: Given the system $y = 2x + 1$ and $y = -x + 4$, the point $(2, 5)$ lies on the first line but not the second line because $5 \neq -2 + 4$. Therefore, it is NOT a solution to the system.