Algebra
Linear functions
Solving linear equations by graphing
20 practice questions
2 video lessons
Theory + worked examples
Theory
To solve an equation by graphing:
- Set each side as a function, \(y=\) left and \(y=\) right.
- Graph both.
- The \(x\)-coordinate of the intersection is the solution.
At the intersection both sides are equal, so the equation is satisfied.
The lines meet at \(x=2\), the solution.
Solving by graphing.
The idea:
\[\text{left}=\text{right}\ \Leftrightarrow\ \text{graphs intersect}\]
Read the \(x\)-coordinate of the intersection point.
How to solve by graphing
- Write each side as \(y=\dots\).
- Graph both functions.
- Find the intersection.
- Read its \(x\)-coordinate and check.
Example 1 — Graph both sides
Solve \(2x-1=3\) by graphing.
Solution
Graph \(y=2x-1\) and \(y=3\); read the intersection.
| \(\text{intersection at}\) | \(x=2\) |
Example 2 — Read the solution
Two lines meet at \((4,7)\). What is the solution?
Solution
The solution is the \(x\)-coordinate.
| \(x\) | \(=\) | \(4\) |
Example 3 — Check by substitution
Check \(x=2\) in \(2x-1=3\).
Solution
Substitute.
| \(2(2)-1\) | \(=\) | \(3\ \checkmark\) |
Example 4 — Why it works
Why does the intersection give the solution?
Solution
At the intersection both sides have the same value, so the equation holds.
Common pitfalls
The solution is the \(x\)-coordinate of the intersection.
Graph accurately or you'll misread the point.
Check by substituting back.
Frequently asked questions
How do you solve an equation by graphing?
Graph both sides and find where they intersect.
What part of the intersection is the solution?
The \(x\)-coordinate.
Why does the intersection give the answer?
Both sides are equal there.
Should you check the graphical solution?
Yes — substitute it back.
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