 # Find the Equation of a Line

In this worksheet, students will find the equations of straight lines when given the line's gradient and a point it passes through, or these can be calculated. They will also apply the similar properties of parallel lines. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs, Straight Line Graphs

Difficulty level:   ### QUESTION 1 of 10

We are able to find the equation of a line when we know its gradient and a point it passes through.

e.g. 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 (i.e. the point where the line crosses the y-axis).

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

This gives us: y = 2x + c

To work out the value of c, we need to substitute the coordinates of the point given to us into our equation.

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

This means that:

5 = 2 × 4 + c

5 = 8 + c

5 - 8 = c

-3 = c

Now let's put everything together to get our final equation for this line: 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.

In this activity, we will find the equations of straight lines when we are given, or can calculate, the line's gradient and a point it passes through.

You will also need to recognise and apply the similar properties of parallel lines.

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 has a gradient of 4 and passes through the point (1, 7).

What is its equation?

Do not use any spaces in your equation between numbers, terms, or symbols or you may be marked incorrectly.

Four lines have a gradient of 5.

They pass through 4 different points.

Find the value of c for each of the lines, given the coordinates of the points shown below.

## Column B

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

A line has a gradient of -2 and passes through the point (-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.

Which is it?

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

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

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

The gradient of the line y = -2x - 1 is -2

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

Complete the blanks below to create an equation which represents this line.

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

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

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

The gradient of the line y = -2x - 1 is -2

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

What is the equation of this line?

Do not use any spaces in your equation between numbers, terms, or symbols or you may be marked incorrectly.

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

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

Pick the correct combination to represent the values of m and c for this line.

A line is parallel to y = 3x + 1 and passes through the point (-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 one represented by the equation: y = -1/2x + 3

It passes through the point (-4, 3).

What is the equation of this 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?

y = 3x + 1
EDDIE SAYS
The gradient of the line is 3, so its equation will be y = 3x + c. Then we need to substitute (1, 4) into the equation so we can find the value of c. 4 = 3 × 1 + c 4 = 3 + c 4 - 3 = c c = 1 If we put this back together, we have found that the final equation of this line is: y = 3x + 1 How did you find this first challenge? Refer back to the Introduction now if you want to brush up on this process before trying another question.
• Question 2

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

What is its equation?

Do not use any spaces in your equation between numbers, terms, or symbols or you may be marked incorrectly.

y=4x+3
y=4x + 3
y = 4x + 3
y = 4x+3
EDDIE SAYS
The gradient of the line is 4. So our starting equation will be y = 4x + c. Now let's substitute the coordinates we know to find the value of c. 7 = 4 × 1 + c 7 = 4 + c 7 - 4 = c c = 3 So our final equation will be: y = 4x + 3
• Question 3

Four lines have a gradient of 5.

They pass through 4 different points.

Find the value of c for each of the lines, given the coordinates of the points shown below.

## 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 We need to substitute the coordinates of each of the points to work out the different possible values of c. Here's an example using (-2, 2): 2 = 5 × -2 + c 2 = -10 + c 2 + 10 = c 12 = c Can you use this example to find the other three matches here independently?
• Question 4

A line has a gradient of -2 and passes through the point (-3, 5).

What is the equation of this line?

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 So our final equation is: y = -2x - 1
• Question 5

One of the statements below is false.

Which is it?

The line y = 3x + 1 passes through the point (3, 1)
EDDIE SAYS
We can find a point of intersection of a line by substituting the coordinates into our equation and checking it works. e.g. y = 2x + 1 and (3, 7) 7 = (2 × 3) + 1 7 = 7 so this statement is true. 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 so this is a false statement We can find the gradient of a line when given its equation using the number representing m in: y = mx + c
• Question 6

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

Complete the blanks below to create an equation which represents this line.

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

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

What is the equation of this line?

Do not use any spaces in your equation between numbers, terms, or symbols or you may be marked incorrectly.

y=-2x+9
y = -2x + 9
y= -2x + 9
y=-2x + 9
y = - 2x + 9
EDDIE SAYS
The gradient of this line is - 2, so our starting equation will be: y = -2x + c Now let's substitute (4, 1) into the equation: 1 = -2 × 4 + c (careful with the negative sign here!) 1 = -8 + c 1 + 8 = c 9 = c So our final equation is: y = -2x + 9
• Question 8

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

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

Pick the correct combination to represent the values of m and c for this line.

EDDIE SAYS
The gradient of this line is 2, so m = 2. Therefore, we also know that: y = 2x + c Now let's substitute (-3, -1) into this equation: -1 = 2 × -3 + c -1 = -6 + c -1 + 6 = c c = 5 Did you select that winning combination?
• Question 9

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

What is the equation of this line?

y = 3x + 11
EDDIE SAYS
Parallel lines have the same gradient, so we know that the gradient of the unknown line is also 3. So y = 3x + c Now we need to substitute the point (-2, 5) into this equation to work out the value of c: 5 = 3 × -2 + c 5 = -6 + c 5 + 6 = c c = 11 So our final equation for this parallel line is: y = 3x + 11
• Question 10

A line is parallel to one represented by the equation: y = -1/2x + 3

It passes through the point (-4, 3).

What is the equation of this line?

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 let's substitute (-4, 3) into this equation to find the value of c: 3 = -1/2 × -4 + c 3 = 2 + c (negative multiplied by a negative is a positive) 3 - 2 = c c = 1 So our final equation of this parallel line is: y = -1/2x + 1 or y = -0.5x + 1 Great job completing this activity! If you found it tricky then why not try our Level 1 activity, or if you are ready for a challenge, our Level 3 activity.
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