Equation from Gradient and Intercept

Jump to section:

Key Ideas Easy Medium Hard

y = mx + c

ax + by + c = 0

What to Remember

Start from slope-intercept form

y = mx + c

Clear the denominator

Multiply by the denominator so every coefficient is an integer.

One side zero

Move every term to one side. The other side is 0.

Neat coefficients

Divide out a common factor. Keep the coefficient of x positive when there is an x term.

Easy

Integer gradient and intercept.

Example

Gradient 3, y-intercept −1. Write ax + by + c = 0.

y = mx + c

y = 3x − 1

Rearrange so one side is 0.

3x − y − 1 = 0

Answer: 3x − y − 1 = 0

Write the equation of the line with gradient 2 and y-intercept 3. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 2x + 3

Rearrange so one side is 0.

2x − y + 3 = 0

Answer: 2x − y + 3 = 0

Write the equation of the line with gradient −1 and y-intercept 4. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −x + 4

Rearrange so one side is 0.

x + y − 4 = 0

Answer: x + y − 4 = 0

Write the equation of the line with gradient 3 and y-intercept −2. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 3x − 2

Rearrange so one side is 0.

3x − y − 2 = 0

Answer: 3x − y − 2 = 0

Write the equation of the line with gradient 1 and y-intercept 0. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = x

Rearrange so one side is 0.

x − y = 0

Answer: x − y = 0

Write the equation of the line with gradient −4 and y-intercept 5. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −4x + 5

Rearrange so one side is 0.

4x + y − 5 = 0

Answer: 4x + y − 5 = 0

Medium

One fraction.

Example

Gradient 23, y-intercept 4. Write ax + by + c = 0.

y = mx + c

y = 23x + 4

Multiply both sides by 3.

3y = 2x + 12

Rearrange so one side is 0.

2x − 3y + 12 = 0

Answer: 2x − 3y + 12 = 0

Write the equation of the line with gradient 12 and y-intercept 3. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 12x + 3

Multiply both sides by 2.

2y = x + 6

Rearrange so one side is 0.

x − 2y + 6 = 0

Answer: x − 2y + 6 = 0

Write the equation of the line with gradient −23 and y-intercept 1. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −23x + 1

Multiply both sides by 3.

3y = −2x + 3

Rearrange so one side is 0.

2x + 3y − 3 = 0

Answer: 2x + 3y − 3 = 0

Write the equation of the line with gradient 34 and y-intercept −2. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 34x − 2

Multiply both sides by 4.

4y = 3x − 8

Rearrange so one side is 0.

3x − 4y − 8 = 0

Answer: 3x − 4y − 8 = 0

Write the equation of the line with gradient 13 and y-intercept 2. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 13x + 2

Multiply both sides by 3.

3y = x + 6

Rearrange so one side is 0.

x − 3y + 6 = 0

Answer: x − 3y + 6 = 0

Write the equation of the line with gradient −12 and y-intercept −4. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −12x − 4

Multiply both sides by 2.

2y = −x − 8

Rearrange so one side is 0.

x + 2y + 8 = 0

Answer: x + 2y + 8 = 0

Hard

A fraction in the gradient and in the intercept.

Example

Gradient −14, y-intercept 32. Write ax + by + c = 0.

y = mx + c

y = −14x + 32

Multiply both sides by 4.

4y = −x + 6

Rearrange so one side is 0.

x + 4y − 6 = 0

Answer: x + 4y − 6 = 0

Write the equation of the line with gradient 25 and y-intercept 35. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 25x + 35

Multiply both sides by 5.

5y = 2x + 3

Rearrange so one side is 0.

2x − 5y + 3 = 0

Answer: 2x − 5y + 3 = 0

Write the equation of the line with gradient −32 and y-intercept 12. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −32x + 12

Multiply both sides by 2.

2y = −3x + 1

Rearrange so one side is 0.

3x + 2y − 1 = 0

Answer: 3x + 2y − 1 = 0

Write the equation of the line with gradient 34 and y-intercept −12. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 34x − 12

Multiply both sides by 4.

4y = 3x − 2

Rearrange so one side is 0.

3x − 4y − 2 = 0

Answer: 3x − 4y − 2 = 0

Write the equation of the line with gradient −23 and y-intercept 53. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = −23x + 53

Multiply both sides by 3.

3y = −2x + 5

Rearrange so one side is 0.

2x + 3y − 5 = 0

Answer: 2x + 3y − 5 = 0

Write the equation of the line with gradient 52 and y-intercept −34. Give the answer in the form ax + by + c = 0.

Show Answer

y = mx + c

y = 52x − 34

Multiply both sides by 4.

4y = 10x − 3

Rearrange so one side is 0.

10x − 4y − 3 = 0

Answer: 10x − 4y − 3 = 0

Want to go over a line together?

Book a Session with Amy