2.1.2 Writing and Graphing Equations

Solve Systems of Linear Equations

In Unit 1, you learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear 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.

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

A linear equation in two variables, such as $2x+y=7$, 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 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.

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.

Example 1

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

Here is how to do this:

$\{\begin{array}{c}\hfill x-y=-1\hfill \\ \hfill 2x-y=-5\hfill \end{array}$

We substitute $x=-2$ and $y=-1$ into both equations.

$\begin{array}{cc}\hfill x-y=-1& \hfill 2x-y=-5\\ \hfill -2-(-1)\stackrel{?}{=}-1& \hfill 2(-2)-(-1)\stackrel{?}{=}-5\\ \hfill -1=-1✓& \hfill -3\ne -5\end{array}$

$(-2,-1)$ does not make both equations true.

$(-2,-1)$ is not a solution.

Example 2

Determine whether the ordered pair $(-4,-3)$ is a solution to the system$\{\begin{array}{c}x-y=-1\hfill \\ 2x-y=-5\hfill \end{array}$.

Here is how to do this:

We substitute $x=-4$ and $y=-3$ into both equations.

$\begin{array}{cc}\hfill x-y=-1& \hfill 2x-y=-5\\ \hfill -4-(-3)\stackrel{?}{=}-1& \hfill 2(-4)-(-3)\stackrel{?}{=}-5\\ \hfill -1=-1✓& \hfill -5=-5✓\end{array}$

$(-4,-3)$ is true for both equations. It is a solution.

Solve a System of Linear Equations by Graphing

The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And by finding what the lines have in common, we’ll find the solution to the system.

Example

Solve the system by graphing $\{\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}$

Step 1 - Graph the first equation.

To graph the first line, write the equation in slope-intercept form. $2x+y=7$ $y=-2x+7$ So, the slope is -2 and the $y$-intercept is 7.

Step 2 - Graph the second equation on the same rectangular coordinate system.

This line may be easier to graph using the intercepts. To find the $x$-intercept, substitute $y=0$ into the equation. $x-2y=6$ $x-2(0)=6$ $x=6$ To find the $y$-intercept, substitute $x=0$ into the equation. $x-2y=6$ $0-2y=6$ $-2y=6$ $y=-3$ So, the intercepts are $(6,0)$ and $(0,-3)$.

Step 3 - Determine whether the lines intersect, are parallel, or are the same line.

Look at the graph of the lines. In this case, the lines intersect.

Step 4- Estimate the solution to the system.

• If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
• If the lines are parallel, the system has no solution.
• If the lines are the same, the system has an infinite number of solutions.

The lines intersect at $(4,-1)$.

Step 5 - Check the solution in both equations.

$\begin{array}{c}2x+y=7\hfill \\ 2(4)+(-1)\stackrel{?}{=}7\hfill \\ 8-1\stackrel{?}{=}7\hfill \\ 7=7✔\hfill \end{array}$

$\begin{array}{c}x-2y=6\hfill \\ 4-2(-1)\stackrel{?}{=}6\hfill \\ 6=6✔\hfill \end{array}$

The steps to use to solve a system of linear equations by graphing are shown here.

Using a Graph to Solve a System

Step 1 - Graph the first equation.

Step 2 - Graph the second equation on the same rectangular coordinate system.

Step 3 - Determine whether the lines intersect.

Step 4 - Estimate the solution to the system.

If the lines intersect, identify the point of intersection. This is the solution to the system.

Step 5 - Check the solution in both equations.

Try it

Try It: Solve a System of Linear Equations by Graphing

Solve the system by graphing$\{\begin{array}{c}x-3y=-3\hfill \\ x+y=5\hfill \end{array}$

Compare your answer:

Here is how to graph the system:

Step 1 - Graph the first equation.

Step 2 - Graph the second equation on the same rectangular coordinate system.

Step 3 - Determine whether the lines intersect. The lines intersect at one point.

Step 4 - Estimate the solution to the system.

• If the lines intersect, identify the point of intersection. This is the solution to the system.

The lines intersect at $(3,2)$.

Step 5 - Check the solution in both equations.

$\begin{array}{c}x-3y=-3\hfill \\ 3-3(2)\stackrel{?}{=}-3\hfill \\ 3-6\stackrel{?}{=}-3\hfill \\ -3=-3✔\hfill \end{array}$

$\begin{array}{c}x+y=5\hfill \\ 3+2\stackrel{?}{=}5\hfill \\ 5=5✔\hfill \end{array}$