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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.

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:
$(2,1).$

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.

Write the equations in slope-intercept form

Graph the lines

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.

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

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.$\begin{cases}y =2x+5 \\ y=0.5x+2 \end{cases}$

$\begin{cases}{\color{#009600}{1}}\stackrel{?}{=}2({\color{#0000FF}{\text{-} 2}})+5 \\ {\color{#009600}{1}}\stackrel{?}{=}0.5({\color{#0000FF}{\text{-} 2}})+2 \end{cases}$

MultiplyMultiply

$\begin{cases}1 \stackrel{?}{=}\text{-} 4+5 \\ 1\stackrel{?}{=}\text{-} 1+2 \end{cases}$

AddTermsAdd terms

$\begin{cases}1=1 \\ 1=1 \end{cases}$

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

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 $14.$

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