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# Solving Systems of Linear Equations Graphically

A system of equations is a set of two or more equations. In this course, linear systems in two equations will be explored. Unlike an equation, where the solution is usually a value or a set of values, the solution to a system of equations is usually an $x$-$y$ point or set of $x$-$y$ points. In this section, the graphical interpretation of a system and its solution will be discussed.
Concept

## System of Linear Equations

To show that equations are part of the same system they're usually written on top of each other with a curly bracket to the left. It's not unusual to add Roman numerals, to be able to refer to the equations individually. $\begin{cases}x+y=3 & \, \text {(I)}\\ x-y=1 & \text {(II)}\end{cases}$ Systems of equations often contain more than one unknown variable, and the solution is the set of coordinates that make all equations true simultaneously. In the example above, the solution is $x=2$ and $y=1.$ These coordinates make the sides equal in both equations. The solution is usually written as a point: $(2,1).$

System of equations can be solved both graphically and algebraically.
Method

## Solving a System of Linear Equations Graphically

To solve a system of linear equations graphically means graphing the lines and identifying the point of intersection.

For example, the following system, $\begin{cases}2y=\text{-} 2x+8 \\ x=y-1, \end{cases}$ can be solved by graphing.

### 1

Write the equations in slope-intercept form
Start by writing the equations in slope-intercept form by solving for $y.$ For the first equation, divide both sides by $2\text{:}$ $2y=\text{-} 2x+8 \quad \Leftrightarrow \quad y=\text{-} x+4.$ The second equation can be rewritten by adding $1$ on both sides and rearranging it.
$x=y-1$
$x+1=y$
$y=x+1$
The second equation, written in slope-intercept form, is $y=x+1,$ meaning that the system can be written as $\begin{cases}y=\text{-} x+4 \\ y=x+1. \end{cases}$

### 2

Graph the lines

Next, graph the lines on the same coordinate plane. Here, the $y$-intercepts are $b_1=4$ and $b_2=1$ the slopes are $m_1=\text{-} 1$ and $m_2=1.$

### 3

Identify the point of intersection

The point where the lines intersect is the solution to the system.

The lines appear to intersect at $(1.5,2.5).$ Thus, this is the solution to the system.

Note that whenever a system of equations is solved graphically, the solution is approximate.
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Exercise

In the coordinate plane, two lines are graphed.

Use the graph to solve the system $\begin{cases}y=2x+5 \\ y=0.5x+2. \end{cases}$

Show Solution
Solution

We can find the solution to the system by identifying the point of intersection.

From the graph, it can be seen that the lines intersect at the point $(\text{-} 2,1).$ Thus, this point is the solution to the system. We can verify this algebraically by substituting $x=\text{-} 2$ and $y=1$ into both equations. We'll know our answer is correct if both statements made are true.
$\begin{cases}y =2x+5 \\ y=0.5x+2 \end{cases}$
$\begin{cases}{\color{#009600}{1}}\stackrel{?}{=}2({\color{#0000FF}{\text{-} 2}})+5 \\ {\color{#009600}{1}}\stackrel{?}{=}0.5({\color{#0000FF}{\text{-} 2}})+2 \end{cases}$
$\begin{cases}1 \stackrel{?}{=}\text{-} 4+5 \\ 1\stackrel{?}{=}\text{-} 1+2 \end{cases}$
$\begin{cases}1=1 \\ 1=1 \end{cases}$
Thus, since $(\text{-} 2,1)$ satisfies both equations, it's the solution to the system.

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Exercise

In a football game, the home team, the Mortal Wombats, defeated the Fearless Seagulls by $13$ points. The total score for both teams was $41.$ What was the final score?

Show Solution
Solution

To begin, we'll use variables to represent the different quantities. Let $w$ be the number of points the Wombats scored and $s$ be the number of points the Seagulls scored. The Wombats scored $13$ more points than the Seagulls. Thus, the difference between $w$ and $s$ can be written as $w=s+13.$ The total amount of points was $41,$ so the sum of $w$ and $s$ is $w+s=41.$ Both of these equations must be true simultaneously, giving us the following system of equations. $\begin{cases}w=s+13 \\ w+s=41 \end{cases}$ We can solve the system by graphing. First, let's write the second equation in slope-intercept form by subtracting $s$ on both sides. $\begin{cases}w=s+13 \\ w=\text{-} s+41 \end{cases}$ Now, we can graph the lines. Since the scores cannot be negative, we only graph the lines for positive values of $s$ and $w.$

Now, we can identify the point of intersection.

The point of intersection is $(14,27).$ This means, the Wombats scored $27$ points and the Seagulls scored $14.$