Since the solution of a system is where the lines cross, one way to find it is to draw both lines carefully, using what you know about slope and intercept.
Why it works
Graphing shows every solution of each equation at once, so the crossing point can be seen. A drawing is only as exact as the grid, though: if the crossing lands between grid lines, you can only estimate it. That is why you always check the point in both equations.
Key idea
To solve by graphing: graph both lines, find where they intersect, write the point as $(x, y)$, and check it in both equations.
Worked examples
Worked example: Read the crossing
Solve by graphing: $y = x + 1$ and $y = -2x + 7$.
- First line: start at $(0, 1)$, then up 1 and right 1 each step.
- Second line: start at $(0, 7)$, then down 2 and right 1 each step.
- They meet at $(2, 3)$. Check: $2 + 1 = 3$ and $-4 + 7 = 3$.
Answer: $(2, 3)$
Worked example: A fractional slope
Solve by graphing: $y = \frac{1}{2}x + 3$ and $y = -x$.
- First line: from $(0, 3)$, up 1 for every 2 to the right (or down 1 for every 2 to the left).
- Second line: through the origin, down 1 for every 1 to the right.
- They meet at $(-2, 2)$. Check: $\frac{1}{2}(-2) + 3 = 2$ and $-(-2) = 2$.
Answer: $(-2, 2)$
Common mistake
Reporting only the $x$-coordinate, as in “the solution is $x = 2$.” A system in $x$ and $y$ needs both values: the solution is the point $(2, 3)$.