Lines can also be written as $Ax + By = C$, called standard form. You get it from slope-intercept form by moving the $x$ term and clearing fractions. It is handy when the numbers are whole, and especially for graphing quickly by intercepts.
Why it works
The $x$-intercept is where the line meets the $x$-axis, and every point on that axis has $y = 0$. Put $y = 0$ into $Ax + By = C$ and the $y$ term disappears, leaving $Ax = C$. Likewise every point on the $y$-axis has $x = 0$, leaving $By = C$. Two points determine a line, so the two intercepts are enough to draw it.
Key idea
For $Ax + By = C$: set $y = 0$ to find the $x$-intercept, and set $x = 0$ to find the $y$-intercept. To convert from $y = mx + b$, clear fractions and move the $x$ term to the left, with $A \ge 0$.
Worked examples
Worked example: Graph by intercepts
Find the intercepts of $3x + 2y = 12$.
- $x$-intercept: $y = 0$ gives $3x = 12$, so $x = 4$: the point $(4, 0)$.
- $y$-intercept: $x = 0$ gives $2y = 12$, so $y = 6$: the point $(0, 6)$.
Answer: $(4, 0)$ and $(0, 6)$.
Worked example: Convert to standard form
Write $y = \frac{2}{3}x - 5$ in standard form.
- Multiply every term by $3$: $3y = 2x - 15$.
- Subtract $2x$ from both sides: $-2x + 3y = -15$.
- Multiply by $-1$ so that $A$ is positive: $2x - 3y = 15$.
Answer: $2x - 3y = 15$
Worked example: And back
Find the slope of $4x + 5y = 20$.
- Solve for $y$: $5y = -4x + 20$, so $y = -\frac{4}{5}x + 4$.
Answer: The slope is $-\frac{4}{5}$ (in general, $-\frac{A}{B}$).
Common mistake
Setting the wrong variable to $0$: to find the $x$-intercept you set $y = 0$, not $x = 0$. Check: an $x$-intercept is written $(a, 0)$, so its $y$-coordinate is the one that is $0$.