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Gradients, intercepts and y = mx + c
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Every straight line can be written as y = mx + c. The two numbers m and c tell you its steepness and where it starts.
y = mx + c
m is the gradient: how far up for every 1 across. A negative gradient slopes downwards.
c is the y-intercept: where the line crosses the y-axis.
y = 4x − 3 has gradient 4 and crosses the y-axis at (0, −3).
Rearrange into this form first: 2y = 6x + 8 becomes y = 3x + 4.
c is the y-intercept: where the line crosses the y-axis.
y = 4x − 3 has gradient 4 and crosses the y-axis at (0, −3).
Rearrange into this form first: 2y = 6x + 8 becomes y = 3x + 4.
Gradient from two points
gradient = change in ychange in x
Through (1, 3) and (4, 12): (12 − 3)(4 − 1) = 93 = 3.
Parallel lines have the same gradient.
Through (1, 3) and (4, 12): (12 − 3)(4 − 1) = 93 = 3.
Parallel lines have the same gradient.
Solving with graphs
Where two lines cross, both equations are true at once. That point solves the simultaneous equations.
y = x + 1 and y = 3x − 3 cross at (2, 3), so x = 2 and y = 3.
y = x + 1 and y = 3x − 3 cross at (2, 3), so x = 2 and y = 3.
Find the equation of the line with gradient 2 that passes through (3, 1).
- y = 2x + c
- Substitute (3, 1): 1 = 6 + c, so c = −5
Answer: y = 2x − 5
In y = mx + c, m is the gradient (change in y ÷ change in x) and c is the y-intercept. Two lines cross at the solution of their simultaneous equations.
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