 # Identify Parallel Lines

In this worksheet, students will identify if lines are parallel by comparing gradients represented by m in the equation y = mx + c. 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

You can identify parallel lines by recognising the fact that they have the same gradient.

e.g. y = 3x + 1 and y = 3x - 7 are parallel, as they both have a gradient of 3.

e.g. y = 3x + 1 and y = 7 - 3x are not parallel. The first line has a gradient of 3, whereas the second has a gradient of - 3.

We need to make sure that the equations of any line are in the form y = mx + c, as this is the only way of finding what the gradient is.

If the equation of the line has a different form, we need to rearrange it to the form y = mx + c, where m represents the gradient of the line.

Let's put this knowledge into practise now.

e.g. Two lines have the equations y = 5x + 2 and 2y + 6 = 10x. Are these two lines parallel?

y = 5x + 2 is already in the correct form, so it can stay as it is.

The gradient of this line is 5.

2y + 6 = 10x is not in the correct form, so we need to rearrange it.

2y = 10x - 6

y = 5x - 3

The gradient of the second line is 5.

As the gradients of both lines are 5, they are parallel.

In this activity, we will identify if lines are parallel by comparing gradients represented by m in the equation y = mx + c.

Select all the equations of the lines parallel to y = 5x - 1.

5y = x + 2

y = 5x + 3

3y = 15x - 9

2y = 5x + 6

Choose which equation from the options below represents a line which is parallel to y = 2x + 3.

2y = 2x + 5

2x + 5y = 1

6y - 12 = 12x

2y = x + 12

Find the matching pairs of equations of parallel lines below.

## Column B

y = 0.5x + 3
y = 7 - 0.5x
3y = 6 - 1.5x
y = 4x + 3
y = 4x + 1
2y = x + 7
x + y = 5
y = 4 - x

One of the statements below is false.

Which one is it?

The gradient of the line represented by 5y - 3 = 4x is 4

The line represented by y = 4x + 7 is parallel to the line 2x - 0.5y = 3

The line represented by 2y + 3x = 3 and the line 2y + 5y = 3 are not parallel

The gradient of the line represented by 2y + 4x = 8 is -2

Tick all the lines from the options below which are parallel to y = 1/3x + 7.

3y - x = 4

6y = 2x - 1

3y + 5 = x + 1

5 + 3y - x = 0

What will the gradient of a line parallel to 7y - 49x = 63 be?

7x

49

7

-7

Here are equations representing four lines:

Line 1: 4y = 3x + 5

Line 2: y = 3x - 2

Line 3: 5y - 15x = 1

Line 4: 3x + y = 1

Which statement from the options below is true about these lines?

Lines 1 and 2 are parallel

Lines 2 and 3 are parallel

Lines 1 and 4 are parallel

Lines 3 and 4 are parallel

Line 1 has an equation 3x + 5y = 17.

Line 2 is parallel to Line 1.

What is the gradient of Line 2?

3x

-3/5

3/5x

3/5

A line with an equation ax + 3y - 11 = 0 is parallel to the line 6y + 2x + 1 = 0.

What is the value of a?

Which of the following lines is not parallel to 4x - 3y = 2?

0 = 8x - 6y + 9

8x - 6y = 7

4x = 3y - 1

3y + 4x = 2

• Question 1

Select all the equations of the lines parallel to y = 5x - 1.

y = 5x + 3
3y = 15x - 9
EDDIE SAYS
If we consider the equation y = 5x - 1, the number next to x represents m in y = mx + c. Therefore, this represents the gradient, which is 5. We need to ensure all the options are in the form y = mx + c so that we can compare the gradients. 5y = x + 2 > y = 1/5x + 2/5 3y = 15x - 9 > y = 5x - 3 2y = 5x + 6 > y = 5/2x + 3 There are two equations here which have a gradient of 5: y = 5x + 3 3y = 15x - 9 > y = 5x - 3 Did you spot those two?
• Question 2

Choose which equation from the options below represents a line which is parallel to y = 2x + 3.

6y - 12 = 12x
EDDIE SAYS
The gradient of the line y = 2x + 3 is 2. We need to rearrange each of the options so they are in the form y = mx + c so that we can compare the gradients. 2y = 2x + 5 > y = x + 2.5 2x + 5y = 1 > 5y = 1 - 2x > y = 1/5 - 2/5x 6y - 12 = 12x > 6y = 12x - 12 > y = 2x - 2 2y = x + 12 > y = 1/2x + 6 Which of these lines has a gradient of 2? Or 2 representing m in the format y = mx + c?
• Question 3

Find the matching pairs of equations of parallel lines below.

## Column B

y = 0.5x + 3
2y = x + 7
3y = 6 - 1.5x
y = 7 - 0.5x
y = 4x + 1
y = 4x + 3
x + y = 5
y = 4 - x
EDDIE SAYS
Some of the equations need to rearranged into the form y = mx + c. 3y = 6 - 1.5x can be rearranged to y = 2 - 0.5x. The gradient of this line is -0.5. Remember about the minus! x + y = 5 can be rearranged to y = 5 - x. The gradient of this line is -1. Again, make sure you include the minus. To find the matches, we need to identity the equations with the same gradients. y = 4x + 1 and y = 4x + 3 have the same gradient, as they both have the same value linked to x. Can you find the other three matches independently?
• Question 4

One of the statements below is false.

Which one is it?

The gradient of the line represented by 5y - 3 = 4x is 4
EDDIE SAYS
Work through each of the statements one at a time, checking if they are true or false. If we rearrange 5y - 3 = 4x, we reach y = 4/5x + 3/5. So the gradient of this line is 4/5, not 4. This statement appears to be false. Why not check the others so you can be 100% confident in this answer?
• Question 5

Tick all the lines from the options below which are parallel to y = 1/3x + 7.

3y - x = 4
3y + 5 = x + 1
5 + 3y - x = 0
EDDIE SAYS
The line y = 1/3x + 7 has a gradient of 1/3. We need to rearrange the options into the form y = mx + c so we can compare them to this. 3y - x = 4 rearranges to y = 1/3x + 4, so it's parallel. 6y = 2x - 1 rearranges to y = 1/3x - 1/6, so it is not parallel. 3y + 5 = x + 1 rearranges to y = 1/3 x - 4/3, so it's also parallel. 5 + 3y - x = 0 rearranges to y = 1/3x - 5/3, so it'll be parallel as well.
• Question 6

What will the gradient of a line parallel to 7y - 49x = 63 be?

7
EDDIE SAYS
Let's rearrange this equation to start. 7y - 49x = 63 7y = 63 + 49x y = 9 + 7x The gradient is shown by the number in front of x, so it's 7. The key here is to rearrange correctly so take your time to review this if you were not correct first time.
• Question 7

Here are equations representing four lines:

Line 1: 4y = 3x + 5

Line 2: y = 3x - 2

Line 3: 5y - 15x = 1

Line 4: 3x + y = 1

Which statement from the options below is true about these lines?

Lines 2 and 3 are parallel
EDDIE SAYS
Work through each statement one at a time, considering if it is true or false. Let's look at the first one together: are lines 1 and 2 parallel? If they are, they will have the same gradient. We need to rearrange the equation for Line 1 into the format y = mx + c so that we can compare it. If we rearrange it, it becomes y = 3/4x +5/4 so its gradient is 3/4. Line 2 (y = 3x - 2) has a gradient of 3, so these do not match. This statement is therefore false. Are lines 2 and 3 parallel? Try rearranging both equations to the form y = mx + c first. If we do this, we find that they have the same gradient (3). This means they are parallel, so this statement is true.
• Question 8

Line 1 has an equation 3x + 5y = 17.

Line 2 is parallel to Line 1.

What is the gradient of Line 2?

-3/5
EDDIE SAYS
Parallel lines have the same gradient, so if we find the gradient of Line 1, this will also be the gradient of Line 2. Did you remember to rearrange the equation? 3x + 5y = 17 5y = 17 - 3x y = 17/5 - 3/5x The gradient is -3/5 (remember to include the sign!). So the gradient of Line 2, which is parallel, must also be -3/5.
• Question 9

A line with an equation ax + 3y - 11 = 0 is parallel to the line 6y + 2x + 1 = 0.

What is the value of a?

1
EDDIE SAYS
Here we need to rearrange both equations and compare them to find the value of m in the form y = mx + c (represented by a here). First, let's rearrange 6y + 2x + 1 = 0: 6y = -2x - 1 y = -1/3 - 1/6 The gradient of this line must be -1/3. Now, let's rearrange ax + 3y - 11 = 0 3y = - ax + 11 y = -a/3x + 11/3 -a/3 must be equal to -1/3. So the value of a is 1.
• Question 10

Which of the following lines is not parallel to 4x - 3y = 2?

3y + 4x = 2
EDDIE SAYS
First let's rearrange 4x - 3y = 2 to put it in the form y = mx + c and find its gradient. 4x - 3y = 2 can be rearranged to y = 4/3x - 1/2 The gradient of this line is 4/3. 3y + 4x = 2 has a gradient of -4/3, so it is not parallel. All the other lines have a gradient of 4/3, so they are parallel. Congratulations on completing this activity!
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