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Find the Equation of a Line

In this worksheet, students will find an equation of a line, given one point and a gradient.

'Find the Equation of a Line' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

We are able to find an equation of a line knowing its gradient and a point it passes through.

 

Let's say we want to find an equation of a line with a gradient 2, passing through the point (4, 5).

 

Start with a general equation of a straight line y = mx + c.

m represents the gradient, c represents the y-intercept.

We know the gradient, so we can replace m with 2.

This gives us y = 2x + c.

 

To work out the value of c, substitute the coordinates of the point given in the question.

The coordinates here are (4, 5), so x = 4 and y = 5.

 

5 = 2 × 4 + c

5 = 8 + c

5 - 8 = c

-3 = c

 

Now let's put everything together to get the equation y = 2x - 3.

 

It is also useful to know that parallel lines have the same gradient. You can use this property of straight lines when the question does not specifically tell you what the gradient is.

A line has a gradient of 3 and passes through the point (1, 4).

What is its equation?

y = 3x + 4

y = 4x + 3

y = 3x + 1

y = x + 3

A line with a gradient of 4 passes through the point (1, 7).

What is the equation of that line?

y = 4x + 3

y = 4x + 7

y = 7x + 4

y = x + 7

Four lines have a gradient of 5. They pass through 4 different points. Find the values of c for each of the lines knowing the coordinates of the point.

Column A

Column B

c = 1
(3, 14)
c = -1
(-2, 2)
c = 3
(2, 11)
c = 12
(1, 8)

The line has a gradient of -2 and passes through (-3, 5). What is the equation of this line?

y = -3x - 5

y = -2x - 1

y = -2x + 5

y = -2x + 7

One of the statements below is false. Pick the false statement.

The gradient of the line y = 3x + 7 is 3.

The line y = 2x + 1 passes through (3, 7).

The line y = 3x + 1 passes through (3, 1).

The gradient of y = -2x - 1 is -2.

A line has a gradient of 1/2 and goes through (6, 7).

Below, complete the equation of this line.

The gradient of the line y = 3x + 7 is 3.

The line y = 2x + 1 passes through (3, 7).

The line y = 3x + 1 passes through (3, 1).

The gradient of y = -2x - 1 is -2.

A line has a gradient -2 and passes through (4, 1).

What is the equation of this line?

y = -2x + 9

y = -2x - 7

y = 4x + 1

y = x + 4

A line passes through (-3, -1) and has a gradient of 2.

It can be written in a form y = mx + c

Pick the correct combination of the values of m and c for the above line.

y = -2x + 9

y = -2x - 7

y = 4x + 1

y = x + 4

A line is parallel to y = 3x + 1 and passes through (-2, 5).

What is the equation of this line?

y = 5x - 2

y = -2x + 5

y = 3x + 11

y = 3x - 1

A line is parallel to y = -1/2x + 3.

It passes through (-4, 3).

What is the equation of the line?

y = -4x + 3

y = 3x - 4

y = -1/2x + 3

y = -1/2x + 1

  • Question 1

A line has a gradient of 3 and passes through the point (1, 4).

What is its equation?

CORRECT ANSWER
y = 3x + 1
EDDIE SAYS
The gradient is 3, so the equation will be y = 3x + c. Substitute (1, 4) into the equation to find the value of c. 4 = 3 × 1 + c 4 = 3 + c 4 - 3 = c c = 1 The equations is y = 3x + 1.
  • Question 2

A line with a gradient of 4 passes through the point (1, 7).

What is the equation of that line?

CORRECT ANSWER
y = 4x + 3
EDDIE SAYS
The gradient is 4. So the equation will be y = 4x + c. Now substitute the coordinates to find the value of c. 7 = 4 × 1 + c 7 = 4 + c 7 - 4 = c c = 3 The equation will be y = 4x + 3.
  • Question 3

Four lines have a gradient of 5. They pass through 4 different points. Find the values of c for each of the lines knowing the coordinates of the point.

CORRECT ANSWER

Column A

Column B

c = 1
(2, 11)
c = -1
(3, 14)
c = 3
(1, 8)
c = 12
(-2, 2)
EDDIE SAYS
Each of the lines will have a form y = 5x + c. Substitute the coordinates of each of the points to work out the value of c. Here's an example: 2 = 5 × -2 + c 2 = -10 + c 2 + 10 = c 12 = c
  • Question 4

The line has a gradient of -2 and passes through (-3, 5). What is the equation of this line?

CORRECT ANSWER
y = -2x - 1
EDDIE SAYS
Do you remember the steps yet? y = -2x + c (gradient is -2) 5 = -2 × (-3) + c (substitute to find the value of c). 5 = 6 + c 5 - 6 = c c = -1 The equation is y = -2x - 1.
  • Question 5

One of the statements below is false. Pick the false statement.

CORRECT ANSWER
The line y = 3x + 1 passes through (3, 1).
EDDIE SAYS
The line y = 3x + 1 passes through (3, 1) is the false statement. If we substitute (3, 1) into the equation, it doesn't work! 1 = 3 × 3 + 1 1 = 9 + 1 1 = 10 is a false statement!
  • Question 6

A line has a gradient of 1/2 and goes through (6, 7).

Below, complete the equation of this line.

CORRECT ANSWER
EDDIE SAYS
The gradient is the number in front of x, so the first blank must contain 1/2 (or 0.5 as these are equivalent). To find what number goes into the next gap, substitute the coordinates of the point into y = 1/2x + c 7 = 1/2 × 6 + c 7 = 3 + c 7 - 3 = c c = 4 So the correct equation is y = 1/2x + 4.
  • Question 7

A line has a gradient -2 and passes through (4, 1).

What is the equation of this line?

CORRECT ANSWER
y = -2x + 9
EDDIE SAYS
The gradient is - 2, this means that the equation will be y = -2x + c. Now substitute (4, 1) into the equation. 1 = -2 × 4 + c (careful with a negative sign here!) 1 = -8 + c 1 + 8 = c 9 = c The equation is y = -2x + 9.
  • Question 8

A line passes through (-3, -1) and has a gradient of 2.

It can be written in a form y = mx + c

Pick the correct combination of the values of m and c for the above line.

CORRECT ANSWER
EDDIE SAYS
The gradient is 2, so m=2 and therefore y=2x + c Now substitute (-3, -1) into this equation: -1 = 2 × -3 + c -1 = -6 + c -1 + 6 = c c = 5
  • Question 9

A line is parallel to y = 3x + 1 and passes through (-2, 5).

What is the equation of this line?

CORRECT ANSWER
y = 3x + 11
EDDIE SAYS
Parallel lines have the same gradient, so the gradient of the line is 3. y = 3x + c You now need to substitute the point (-2, 5) into the equation to work out the value of c. 5 = 3 × -2 + c 5 = -6 + c 5 + 6 = c c = 11 The equation is y = 3x + 11.
  • Question 10

A line is parallel to y = -1/2x + 3.

It passes through (-4, 3).

What is the equation of the line?

CORRECT ANSWER
y = -1/2x + 1
EDDIE SAYS
Did you remember that parallel lines have the same gradient? From this, we know that y = -1/2x + c. Now substitute (-4, 3) into the equation to find the value of c and complete the equation. 3 = -1/2 × -4 + c 3 = 2 + c (negative multiplied by a negative is a positive) 3 - 2 = c c = 1 The equation of the line is y = -1/2x + 1
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