Solving Systems of Equations by Graphing

A system of equations is just two or more equations that you're trying to solve at the same time. For a system of two linear equations, you're looking for a point that satisfies both equations—a point that makes both equations true.

Graphically, this means finding where the two lines intersect. The intersection point's coordinates are the solution to the system.

What Does a Solution Look Like?

A solution to a system is an ordered pair $(x, y)$ that makes both equations true.

Example: Is $(2, 3)$ a solution to this system?

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

Check the first equation: $3 = 2 + 1 = 3$. True.

Check the second equation: $3 = 2(2) - 1 = 4 - 1 = 3$. True.

Yes, $(2, 3)$ satisfies both equations, so it's the solution.

Solving by Graphing

The graphical method is straightforward: graph both lines on the same coordinate plane, then find where they intersect.

Example: Solve this system by graphing.

$$\begin{cases} y = 2x - 1 \\ y = -x + 5\end{cases}$$

Both equations are in slope-intercept form, making them easy to graph.

For $y = 2x - 1$: $y$-intercept is -1, slope is 2. Plot $(0, -1)$, then go up 2 and right 1 to get $(1, 1)$. Draw the line.

For $y = -x + 5$: $y$-intercept is 5, slope is -1. Plot $(0, 5)$, then go down 1 and right 1 to get $(1, 4)$. Draw the line.

The lines intersect at $(2, 3)$. That's the solution.

Verify: $3 = 2(2) - 1 = 3$ ✓ and $3 = -2 + 5 = 3$ ✓

Example: Solve by graphing.

$$\begin{cases} x + y = 6 \\ 2x - y = 3\end{cases}$$

These aren't in slope-intercept form, but we can either convert them or just find a couple points for each.

For $x + y = 6$: when $x = 0$, $y = 6$. When $x = 3$, $y = 3$. Plot $(0, 6)$ and $(3, 3)$.

For $2x - y = 3$: when $x = 0$, $-y = 3$, so $y = -3$. When $x = 3$, $6 - y = 3$, so $y = 3$. Plot $(0, -3)$ and $(3, 3)$.

Both lines pass through $(3, 3)$. That's the solution.

Three Types of Systems

Systems of two linear equations fall into three categories based on how many solutions they have.

One solution (consistent and independent): The lines intersect at exactly one point. Most systems fall into this category. The lines have different slopes.

No solution (inconsistent): The lines are parallel and never intersect. This happens when the lines have the same slope but different $y$-intercepts.

Example: $$\begin{cases} y = 2x + 1 \\ y = 2x - 3\end{cases}$$

Both lines have slope 2. The first has $y$-intercept 1, the second has $y$-intercept -3. They're parallel, so there's no solution.

Infinitely many solutions (consistent and dependent): The two equations represent the same line. Every point on the line is a solution.

Example: $$\begin{cases} y = 3x + 2 \\ 2y = 6x + 4\end{cases}$$

The second equation is just the first multiplied by 2. If you solve the second for $y$, you get $y = 3x + 2$, which is identical to the first. The lines are the same, so infinitely many solutions.

Limitations of Graphing

Graphing is great for visualization and understanding what a system means geometrically. However, it has limitations:

If the solution involves fractions or decimals that aren't nice numbers, it's hard to read the exact coordinates from a graph. The solution might be $(2.7, 3.4)$, and you can only estimate that visually.

If the intersection point is far from the origin, you might need a large graph or have to zoom out, making precision difficult.

For these reasons, algebraic methods (substitution and elimination, which you'll learn next) are often more accurate and efficient for finding exact solutions.

Steps for Solving by Graphing

Convert both equations to a form you can easily graph (slope-intercept is usually easiest).
Graph the first equation carefully, using at least two points.
Graph the second equation on the same axes.
Identify the point where the lines intersect. That's your solution.
Check your answer by substituting the coordinates into both original equations.

Test Your Skills

Solve each system by graphing:

$$\begin{cases} y = x + 2 \\ y = -x + 4\end{cases}$$

$$\begin{cases} y = 2x - 1 \\ x + y = 5\end{cases}$$

$$\begin{cases} y = 3x \\ y = 3x + 5\end{cases}$$

$$\begin{cases} x + 2y = 8 \\ 3x - 2y = 0\end{cases}$$

Answers:

Graph both lines. $y = x + 2$ has slope 1, $y$-intercept 2. $y = -x + 4$ has slope -1, $y$-intercept 4. They intersect at $(1, 3)$.
$y = 2x - 1$ and $y = -x + 5$ (from converting the second equation). They intersect at $(2, 3)$.
Both lines have slope 3 but different $y$-intercepts. They're parallel, so no solution.
Converting: $y = -\frac{1}{2}x + 4$ and $y = \frac{3}{2}x$. They intersect at $(2, 3)$.

Where Errors Happen

Drawing inaccurate graphs is the root of most wrong answers with this method. Use a ruler, plot your points carefully, and make your graph large enough to read precisely. Sloppy graphing leads to sloppy solutions.

Misreading the intersection point is deceptively easy. Even experienced students sometimes read $(3, 2)$ as $(2, 3)$. Double-check by substituting your answer into both equations to verify it works.

Assuming all systems have exactly one solution will bite you eventually. Some systems have no solution (parallel lines that never meet), and some have infinitely many solutions (the same line written two ways). Look at the slopes before you even start graphing to get a sense of what to expect.

Not checking your answer is tempting when you're confident, but it's also when you're most likely to have made a small error. Plug your solution into both original equations. It takes thirty seconds and prevents embarrassing mistakes.

Forgetting to extend your lines far enough sometimes means the intersection isn't visible on your initial graph. If you don't see where the lines cross, expand your coordinate plane or extend the lines further.