Solving Systems by Graphing

A system of equations is a set of two or more equations that you deal with all together at once. When you solve a system of linear equations by graphing, you are looking for the exact point where the two lines cross.

The coordinates of this intersection point $(x, y)$ represent the solution, because it is the only point that makes both equations true at the same time.

Steps to Solve by Graphing

1. Graph the first equation: Plot the y-intercept and use the slope to draw the line.
2. Graph the second equation: Draw this line on the exact same coordinate plane.
3. Find the intersection: Identify the $(x, y)$ coordinates where the two lines cross.
4. Check your work: Plug the $x$ and $y$ values back into both original equations to verify they are correct.

Example 1

Solve by graphing: $y = x + 1$ and $y = -x + 3$

• Line 1 ($y = x + 1$): The y-intercept is $1$ and the slope is $1$.
• Line 2 ($y = -x + 3$): The y-intercept is $3$ and the slope is $-1$.

When you graph both lines, they intersect at the point $(1, 2)$.

Check:

• For $y = x + 1$: Does $2 = 1 + 1$? Yes!
• For $y = -x + 3$: Does $2 = -(1) + 3$? Yes!

The solution is $x = 1$, $y = 2$.

Example 2

Solve by graphing: $y = 2x - 1$ and $y = x + 2$

• Line 1 ($y = 2x - 1$): The y-intercept is $-1$ and the slope is $2$.
• Line 2 ($y = x + 2$): The y-intercept is $2$ and the slope is $1$.

Graphing these lines reveals that they cross at the point $(3, 5)$.

Check:

• For $y = 2x - 1$: Does $5 = 2(3) - 1$? Yes ($5 = 6 - 1$).
• For $y = x + 2$: Does $5 = 3 + 2$? Yes.

The solution is $x = 3$, $y = 5$.

Special Cases

Not all systems have exactly one solution:

• No Solution: If the lines are parallel (they have the same slope but different y-intercepts), they will never intersect.
• Infinitely Many Solutions: If both equations graph the exact same line, every point on the line is a valid solution.