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9.4 Y = Mx + C

GCSE Maths Foundation · Edexcel · Chapter 9: Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: gradient and y-intercept, finding the equation of a line, parallel lines and where lines meet. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Read the gradient and y-intercept from an equation y = mx + c
  • Rearrange an equation into the form y = mx + c
  • Find the equation of a line from a graph, from two points, or from a point and a gradient
  • Use gradients to spot parallel lines, and use graphs to find where two lines meet

Key words

  • y-intercept: The point where a line crosses the y-axis. In y = mx + c it is (0, c).
  • Gradient m: The number in front of x in y = mx + c. It tells you how steep the line is.
  • Coefficient: The number multiplying a letter, e.g. in 5x the coefficient of x is 5.
  • Point of intersection: The point where two lines cross. It lies on both lines.

The equation y = mx + c

Every straight line (except a vertical one) can be written as y = mx + c. The value of m is the gradient. The value of c is the y-intercept: the line crosses the y-axis at (0, c).

  • y = 3x + 2; Written as y = mx + c y = 3x + 2; Gradient m 3; y-intercept c 2
  • y = 5 − x; Written as y = mx + c y = −1x + 5; Gradient m −1; y-intercept c 5
  • y = x/2 − 4; Written as y = mx + c y = (1/2)x − 4; Gradient m 1/2; y-intercept c −4
  • y = 7x; Written as y = mx + c y = 7x + 0; Gradient m 7; y-intercept c 0 (through the origin)

Key fact: Look at the sign in front of each term. In y = 5 − 2x the gradient is −2, not 5.

Rearranging into y = mx + c

If the equation does not start “y =”, rearrange it. Do the same to both sides, and divide every term.

Worked example: dividing every term

Find the gradient and y-intercept of 2y = 6x + 8.
Divide every term by 2: y = 3x + 4.

Answer: gradient 3, y-intercept 4

Worked example: moving a term

Find the gradient and y-intercept of x + y = 5.
Subtract x from both sides: y = −x + 5.

Answer: gradient −1, y-intercept 5

The equation of a line from a graph

Read c where the line crosses the y-axis. Then draw a triangle to find the gradient m. Put both into y = mx + c.

Worked example: from a graph

The line crosses the y-axis at (0, −3), so c = −3.
From (1, −1) to (3, 3) it goes up 4 and across 2, so m = 4 ÷ 2 = 2.

Answer: y = 2x − 3

The equation from points and gradients

If you know m and one point, substitute the point into y = mx + c to find c. If you know two points, find m first.

Worked example: a point and a gradient

A line has gradient 3 and passes through (2, 5). Find its equation.
y = 3x + c. Substitute x = 2, y = 5: 5 = 3 × 2 + c, so 5 = 6 + c and c = −1.

Answer: y = 3x − 1

Worked example: two points

Find the equation of the line through (1, 4) and (3, 10).
m = (10 − 4) ÷ (3 − 1) = 6 ÷ 2 = 3. Then 4 = 3 × 1 + c, so c = 1.

Answer: y = 3x + 1

Worked example: a parallel line

Find the equation of the line parallel to y = 2x + 7 that passes through (3, 1).
Parallel, so m = 2. Then 1 = 2 × 3 + c, so c = 1 − 6 = −5.

Answer: y = 2x − 5

Where two lines meet

Draw both lines accurately on the same grid. The point where they cross is the only point on both lines, so its coordinates fit both equations.

Worked example: checking an intersection

The graphs suggest the lines meet at (2, 3). Check: 2 + 1 = 3 ✓ and 7 − 2 × 2 = 3 ✓.

Answer: (2, 3)

Common mistakes

  • Giving the x-intercept as c. The y-intercept is where the line crosses the y-axis.
  • Reading the gradient of y = 5 − 2x as 5. The gradient is the coefficient of x: −2.
  • Only dividing one term when rearranging, e.g. 2y = 6x + 8 → y = 3x + 8. Divide every term.
  • Writing the answer as “m = 3, c = 1” when asked for the equation. Write y = 3x + 1.

Revision questions

Try these, then check your answers below each one.

Question 1. Write down the gradient of the line y = 6 - 4x.

Answer: -4

Working: y = -4x + 6, so the gradient is -4.

Question 2. Find the y-intercept of the line 2y = 8x - 10.

Answer: -5

Working: y = 4x - 5, so the y-intercept is -5.

Question 3. A line has gradient 2 and crosses the y-axis at (0, −3). What is its equation? (Choose from: y = 2x - 3, y = -3x + 2, y = 2x + 3, y = 3x - 2.)

Answer: y = 2x - 3

Working: m = 2, c = -3: y = 2x - 3.

Related lessons

Continue in the full course

This lesson is section 9.4 (EGCMF0904) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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