Skill: Interpret intersections as solutions to systems

An intersection is a point that lies on both graphs. Therefore, its coordinates make both equations true, so the intersection is a solution to the system.

Example

Find the solution to the system by interpreting the graphs:

$$\begin{cases} y=2x+1\\ y=-x+7 \end{cases}$$

Step 1: Recognize the two graphs

Both equations are in slope-intercept form, $y=mx+b$.

• For $y=2x+1$, the slope is $2$ and the $y$-intercept is $1$.
• For $y=-x+7$, the slope is $-1$ and the $y$-intercept is $7$.

Graph both lines on the same coordinate plane.

Step 2: Identify the intersection

The two lines cross at the point

$$(2,5).$$

This means the graphical solution to the system is

$$\boxed{(2,5)}.$$

Step 3: Check that the point works in both equations

Substitute $x=2$ and $y=5$ into the first equation:

$$5=2(2)+1$$ $$5=5$$

Now check the second equation:

$$5=-2+7$$ $$5=5$$

The point $(2,5)$ makes both equations true, so it is the solution.

For a system of equations, the graphs can have:

• one intersection: one solution;
• no intersection: no solution;
• the same graph: infinitely many solutions.

For a system of inequalities, the solution is the region where the shaded parts overlap. A solid boundary is included, while a dashed boundary is not.