Algebra
Linear functions
Writing linear equations
20 practice questions
2 video lessons
Theory + worked examples
Theory
To write a linear equation, use what you're given:
- Slope + point: use point-slope form.
- Two points: find the slope first, then point-slope.
- Slope + \(y\)-intercept: write \(y=mx+b\) directly.
- Graph / table: read \(b\) and the slope.
You always need the slope and one point (or the \(y\)-intercept).
Choose a method from what you're given.
Writing a line from two points.
The essentials:
\[m=\dfrac{y_2-y_1}{x_2-x_1},\qquad y-y_1=m(x-x_1)\]
Find the slope first when given two points.
How to write a linear equation
- Find the slope (from two points if needed).
- Use a known point in point-slope form.
- Or use \(y=mx+b\) if you have the \(y\)-intercept.
- Simplify to the requested form.
Example 1 — Slope and point
Write the line with slope \(2\) through \((0,3)\).
Solution
The point is the \(y\)-intercept.
| \(y\) | \(=\) | \(2x+3\) |
Example 2 — Two points
Write the line through \((1,2)\) and \((3,8)\).
Solution
Find \(m\), then use point-slope.
| \(m\) | \(=\) | \(\dfrac{8-2}{3-1}=3\) |
| \(y-2\) | \(=\) | \(3(x-1)\) |
| \(y\) | \(=\) | \(3x-1\) |
Example 3 — Slope and y-intercept
Write the line with slope \(-1\) and \(y\)-intercept \(4\).
Solution
Use \(y=mx+b\).
| \(y\) | \(=\) | \(-x+4\) |
Example 4 — From a table
A table shows \((0,5),(1,7),(2,9)\). Write the equation.
Solution
The \(y\)-intercept is \(5\), slope \(2\).
| \(y\) | \(=\) | \(2x+5\) |
Common pitfalls
Find the slope first when given two points.
The \(y\)-intercept is \(b\), the point at \(x=0\).
Simplify to the form asked for.
Frequently asked questions
How do you write a line from two points?
Find the slope, then use point-slope form.
What do you need to write a line?
The slope and one point (or the \(y\)-intercept).
How do you get the equation from a graph?
Read the \(y\)-intercept and the slope.
Which form is easiest from slope and y-intercept?
Slope-intercept, \(y=mx+b\).
More in Linear functions