MathematicsExam code: 0580

Linear Graphs

Coordinate Geometry and Graphs

Any straight line that is not vertical can be written y = mx + c: the number m sets its steepness and the number c fixes where it meets the y-axis
The line therefore passes through (0, c)
- y = 4x − 9 has gradient 4 and meets the y-axis at −9
- y = 11 − 2x has gradient −2 and meets the y-axis at 11, so take the number attached to x, not the one written first
A horizontal line has the form y = c, since every point on it has the same y value
A vertical line has the form x = k, and it cannot be written as y = mx + c at all

These papers also use the form ax + by = c, which has to be rearranged before the gradient can be read
Get a single y on one side, then divide every term by whatever multiplies it
- 3x + 4y = 24 becomes 4y = −3x + 24, then y = −0.75x + 6, so the gradient is −0.75 and it crosses the y-axis at 6
Leaving 4y on the left and reading the gradient as −3 is the standard error here
Answers must be given fully simplified, so tidy fractions and collect any like terms before writing the final line

Worked example

Reading a line from its equation

Write down the gradient and the y-axis crossing of y = 3x − 4, and of 2x + 5y = 20.

Solution:

The first is already in the form y = mx + c
The number in front of x is the gradient, so the gradient is 3
The constant is where the line crosses the y-axis, at (0, −4)
The second is not in that form, so rearrange it before reading anything
Subtract 2x: 5y = −2x + 20
Divide every term by 5: y = −0.4x + 4
The gradient is −0.4 and the line crosses the y-axis at (0, 4)
Reading a gradient of 2 straight off 2x + 5y = 20 is the standard slip: nothing can be read until y stands alone

Worked example

Where a line meets a horizontal line

The lines y = 25 − 4x and y = −3 cross at the point P. Find the coordinates of P.

Solution:

Every point on y = −3 has a y-coordinate of −3, so P does too
Substitute y = −3 into the other equation: −3 = 25 − 4x
Subtract 25 from both sides: −28 = −4x
Divide both sides by −4: x = 7
P is (7, −3)
Check in y = 25 − 4x: 25 − 4 × 7 = 25 − 28 = −3
Give both coordinates when the point is asked for: the −3 comes straight from the horizontal line

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