Two equations, two lines, one question: where do they meet? Solving a system of equations by graphing answers that question with a picture. You draw each equation as a line on the same grid, and the crossing point is the solution, because it’s the only point that lies on both lines. Graphing is rarely the fastest way to get an exact answer, but it’s the best way to see a system, and it’s often the method a test asks for by name.
How to Solve a System of Equations by Graphing
- Put each equation in a graphable form. Slope-intercept form,
, is the easiest. For standard form, , plotting the intercepts is quicker. - Graph the first line. Plot the
-intercept, then use the slope (rise over run) to plot a second point, and draw the line through them. - Graph the second line on the same axes, the same way.
- Find the intersection. Read the coordinates of the point where the lines cross.
- Check the point in both original equations.
Behind the picture there’s a simple equation. Two lines
The graph shows you where that happens; the algebra confirms it.
Example 1: Graphing From Slope-Intercept Form
Solve by graphing:
Step 1 — graph
Step 2 — graph
Step 3 — read the intersection: the lines cross at
Step 4 — check:
Answer:
Example 2: Graphing From Standard Form
Solve by graphing:
Both equations are in standard form, so use their intercepts rather than rearranging.
Step 1 — intercepts of
Step 2 — intercepts of
Step 3 — read the intersection:
Step 4 — check:
Answer:
Reading the Number of Solutions From a Graph
The graph also tells you what kind of answer to expect, before any algebra:
| What you see | Number of solutions |
|---|---|
| the lines cross at one point | exactly one |
| the lines are parallel | none |
| the lines lie on top of each other | infinitely many |
You can spot the last two before drawing anything: parallel lines share a slope but have different intercepts, and identical lines share both. The systems of linear equations lesson explains why these are the only three possibilities.
When Graphing Only Gives an Estimate
Graphing is exact only when the lines meet at a point you can read precisely. Solve
The best you can say from the picture is “about
This is the honest limit of the graphing method. Use it to see the solution and estimate it, then use the substitution method or the elimination method when you need exact values.
Choose the System of Equations That Matches a Graph
A common question runs the other way: given two graphed lines, which system do they represent? Read each line’s equation straight off the picture.
Step 1 — the blue line: it crosses the
Step 2 — the green line: it crosses at
Step 3 — the system and its solution:
Step 4 — confirm the intersection:
In multiple-choice versions, the answer choices may be written in standard form. Rearrange each to
Graphing Systems With Technology
On a graphing calculator, enter each equation as
Worked Example: Comparing Two Taxi Fares
Taxi A charges
Step 1 — write the system, with
Step 2 — graph Taxi A from
Step 3 — read the intersection: the lines cross at
Step 4 — check:
Both taxis charge
Common Mistakes to Avoid
- Using different scales on the two axes without noticing. Slopes look wrong and the crossing point gets misread. Label your scale.
- Plotting the slope backwards. A slope of
means down , right , not right , down . - Trusting an estimate as exact. If the crossing isn’t on a grid point, write “approximately” and confirm algebraically.
- Skipping the check. A point read from a sketch is a guess until it satisfies both equations.
- Drawing short line segments. Extend both lines across the grid; a crossing just off the end of a short segment is easy to miss.
Practice Problems
Graph each system (or picture it), then confirm your answer algebraically.
Problem 1. Solve by graphing:
Show answer
Step 1 — graph: the first line starts at
Step 2 — the lines cross at
Answer:
Problem 2. Solve by graphing:
Show answer
Step 1 — graph from intercepts
Step 2 — the lines cross at
Answer:
Problem 3. Solve by graphing:
Show answer
Step 1 — intercepts:
Step 2 — the lines cross at
Answer:
Problem 4. Solve by graphing:
Show answer
Step 1 — compare: both lines have slope
Step 2 — the lines are parallel and never cross.
Answer: no solution
Problem 5. One line has
Show answer
Step 1 — first line:
Step 2 — second line: slope
Step 3 — set them equal:
Answer:
Problem 6. Estimate the solution of
Show answer
Step 1 — estimate: the lines cross a little to the right of
Step 2 — solve exactly:
Answer: about
Graphing makes systems visual, and that habit pays off. When one of the equations isn’t a straight line, the same picture-first approach is the clearest way to see how many solutions to expect.