Interpret intersections as solutions to systems

An intersection of two graphs represents an ordered pair that makes both equations true simultaneously, so solving a system graphically means identifying all common points. For inequalities, the solution is the overlap of the corresponding regions, with boundary lines included for non-strict inequalities and excluded for strict ones; systems may have one, none, or infinitely many solutions. The focus is on two-variable systems represented algebraically and graphically, not higher-dimensional, parametric, or advanced nonlinear systems.

Detailed Explanation: 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.

Learn by doing: Interpret intersections as solutions to systems

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Types of Solutions - Graph to Zero, One, or Infinite

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