# 5.1 Solve systems of equations by graphing

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By the end of this section, you will be able to:
• Determine whether an ordered pair is a solution of a system of equations
• Solve a system of linear equations by graphing
• Determine the number of solutions of linear system
• Solve applications of systems of equations by graphing

Before you get started, take this readiness quiz.

1. For the equation $y=\frac{2}{3}x-4$
is $\left(6,0\right)$ a solution? is $\left(-3,-2\right)$ a solution?
If you missed this problem, review [link] .
2. Find the slope and y -intercept of the line $3x-y=12$ .
If you missed this problem, review [link] .
3. Find the x - and y -intercepts of the line $2x-3y=12$ .
If you missed this problem, review [link] .

## Determine whether an ordered pair is a solution of a system of equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.

Now we will work with systems of linear equations , two or more linear equations grouped together.

## System of linear equations

When two or more linear equations are grouped together, they form a system of linear equations.

We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

$\left\{\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}$

A linear equation in two variables, like 2 x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x , y ) that make both equations true. These are called the solutions to a system of equations .

## Solutions of a system of 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.

Let’s consider the system below:

$\left\{\begin{array}{c}3x-y=7\hfill \\ x-2y=4\hfill \end{array}$

Is the ordered pair $\left(2,-1\right)$ a solution? The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

Let’s try another ordered pair. Is the ordered pair (3, 2) a solution? The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

Determine whether the ordered pair is a solution to the system: $\left\{\begin{array}{c}x-y=-1\hfill \\ 2x-y=-5\hfill \end{array}$

$\left(-2,-1\right)$ $\left(-4,-3\right)$

## Solution

1. (−2, −1) does not make both equations true. (−2, −1) is not a solution. (−4, −3) does not make both equations true. (−4, −3) is a solution.

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