The equation of a straight line
The equation of a straight line
- Most lines are $y = mx + c$: $m$ is the gradient, $c$ is the $y$-intercept.
- Given as $ax + by = c$? Rearrange into $y = mx + c$ to read off both.
- Example: $5x + 4y = 8 \Rightarrow 4y = -5x + 8 \Rightarrow y = -\tfrac54 x + 2$ (gradient $-\tfrac54$, intercept $2$).
Practice
Rearrange 5x + 4y = 8 into y = mx + c. What is c (the y-intercept)?
4y = −5x + 8, so y = −(5/4)x + 2; the intercept is 2.
Practice
In y = mx + c, the letter m stands for the:
m is the gradient (steepness); c is the y-intercept.
Finding the equation
- Know the gradient $m$ and one point? Put the point into $y = mx + c$ to find $c$.
- Example: gradient $3$ through $(1, 2)$: $\;2 = 3(1) + c \Rightarrow c = -1$, so $y = 3x - 1$.
- Given two points? Find the gradient first, then do the same.
Practice
A line has gradient 3 and passes through (1, 2). In y = 3x + c, what is c?
2 = 3(1) + c, so c = 2 − 3 = −1; the line is y = 3x − 1.
You've got it
Key idea
- straight line: $y = mx + c$ ($m$ = gradient, $c$ = $y$-intercept)
- rearrange $ax + by = c$ into $y = mx + c$ to read $m$ and $c$
- gradient $3$ through $(1,2)$ → $y = 3x - 1$