Equation of a line, y=a(x−p)+q

A line with slope $a$ through the point $(p,\,q)$ is $y=a(x-p)+q$. Expanding gives $y=ax+(q-ap)$, the $y=ax+b$ form, where $b$ is the $y$-intercept.

Given two points $(x_1,\,y_1)$ and $(x_2,\,y_2)$, first find the slope, then substitute either point as $(p,\,q)$.

The tangent line $y=f'(a)(x-a)+f(a)$ is exactly this form, with slope $f'(a)$ through $(a,\,f(a))$.

Examples

1. 01Slope 2, through (0, −2). What is the y-intercept?$\displaystyle$AnswerHide

   Answer$\displaystyle -2$
2. 02The line through these two points has slope -2. Write it as y=ax+b:$\displaystyle (0,\ -1),\quad (2,\ -5)$AnswerHide

   Answer$\displaystyle y=-2x-1$
3. 03The line through these two points has slope -3. Write it as y=ax+b:$\displaystyle (-3,\ 0),\quad (-1,\ -6)$AnswerHide

   Answer$\displaystyle y=-3x-9$
4. 04The line through these two points has slope 3. Write it as y=ax+b:$\displaystyle (1,\ -3),\quad (2,\ 0)$AnswerHide

   Answer$\displaystyle y=3x-6$
5. 05The line through these two points has slope 2. Write it as y=ax+b:$\displaystyle (2,\ 2),\quad (5,\ 8)$AnswerHide

   Answer$\displaystyle y=2x-2$
6. 06The line through these two points has slope 3. Write it as y=ax+b:$\displaystyle (-2,\ 1),\quad (1,\ 10)$AnswerHide

   Answer$\displaystyle y=3x+7$

FAQ

Q1When should I use y=ax+b versus y=a(x−p)+q?Show answerHide

A

They describe the same line. $y=a(x-p)+q$ shows the slope $a$ and a point $(p,q)$ directly; $y=ax+b$ shows the y-intercept $b$. Expand to convert: $b=q-ap$.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05