Mathematics (Core)Exam code: 0580
Linear Graphs
Coordinate Geometry and Graphs
Reading a line from its equation
- 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
- 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.
Solution:
- The equation 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)
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