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Solve Equations with Unknown on Both Sides

In this worksheet, students will learn how to solve linear equations with unknowns on both sides.

'Solve Equations with Unknown on Both Sides' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

'If I think of a number, multiply it by 10 and then add 2 it is the same as if I multiply it by 7 and add 14. What number am I thinking of?

Puzzles like this one have been used before in the other 'Equations' activities, but this one is more complicated. The answer is not just a number but I have done something else to the original number. This type of puzzle can still be written as an equation, but now we have to use an x (or whatever letter we use to stand for the number) on each side of the equal sign. So, the equation for this puzzle looks like this:

10x + 2 = 7x + 14

This looks more complicated but it can still be solved using the 'balancing' method we have used before, we just have to add an extra step. The first step is to change it into an equation where the 'x' is only on one side. There are 10 x's on the left and 7 x's on the right. If I remove the 7 x's from the right there will be no x's there. But, remember whatever I do to the right I must do to the left as well. This is how we set it out:

Now, the equation is just the same as we have solved before so you should know what to do. We need to subtract 2 from both sides, then divide by 3.The full working looks like this.

Hope you're following this. Here's another one to help.

3x - 5 = 10 - 2x

Now this time you will notice on the right-hand side the 2x is subtracted from 10. So, what we do in this case is add 2 x's to both sides first. This will get rid of the -2x. Here is the full working:

The rule to remember with these equations is look for the side with the fewest x's on and take them from both sides. If the x has a '-' sign in front then add them on instead of taking them away. Does it make sense? Have a go at the questions and don't worry if you get them wrong, just look at the solution.

When solving the equation

7x - 3 = 4x + 12

which of the following is the best first step? (you do not need to solve the equation)

Add 4x to both sides.

Subtract 4x from both sides.

Add 7x to both sides.

Subtract 7x from both sides.

When solving the equation

2p - 8 = 12 - 3p

which of the following is the best first step? (You do not need to solve the equation.)

Add 2p to both sides.

Subtract 2p from both sides.

Add 3p to both sides.

Subtract 3p from both sides.

There are three steps to solving the following equation

8x - 22 = 2x + 8

The steps are given below. Can you put them in the correct order? 

Column A

Column B

1st
Add 22 to both sides.
2nd
Divide both sides by 6.
3rd
Subtract 2x from both sides.

There are three steps to solving the following equation

23 - 7y = 35 - 13y

The steps are given below. Can you put them in the correct order? 

Column A

Column B

1st
Add 13x to both sides.
2nd
Divide both sides by 6.
3rd
Subtract 23 from both sides.

Which of the following is the correct solution to the equation

2x + 3 = 5x - 2?

x = 3/5

x = 5/3

x = 2

x = 5

When solving the equation 

20 - 7x = 6x - 6

which of the following is the correct solution?

 

x = 1

x = 6/7

x = 7/6

x = 2

Match the following equations to their solutions.

 

Column A

Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

This equation has brackets on the right-hand side as well as x's on both sides. Expand (multiply out) the brackets, then solve as usual. Fill your solution in the space.

4(x - 2) = 3x + 5

Column A

Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

Here's another equation with brackets, this time on both sides. Expand both sets of brackets, then proceed as usual. Fill in your solution below.

5(2 - p) = 2(4 - 2p)

Column A

Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

Solve the following equation by expanding the brackets first. Fill in your solution in the space leaving it as a fraction.

3(4y + 3) = 2(3y + 5) 

 

Column A

Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4
  • Question 1

When solving the equation

7x - 3 = 4x + 12

which of the following is the best first step? (you do not need to solve the equation)

CORRECT ANSWER
Subtract 4x from both sides.
EDDIE SAYS
The smallest number of x's is 4x on the right-hand side so we subtract 4x from both sides.
  • Question 2

When solving the equation

2p - 8 = 12 - 3p

which of the following is the best first step? (You do not need to solve the equation.)

CORRECT ANSWER
Add 3p to both sides.
EDDIE SAYS
The smallest number of p's is -3p on the right-hand side (remember -3p is smaller than 2p). So, we must add 3p to both sides. Did you remember to add because the sign in front of the 3p is negative?
  • Question 3

There are three steps to solving the following equation

8x - 22 = 2x + 8

The steps are given below. Can you put them in the correct order? 

CORRECT ANSWER

Column A

Column B

1st
Subtract 2x from both sides.
2nd
Add 22 to both sides.
3rd
Divide both sides by 6.
EDDIE SAYS
The smallest number of x's is 2x on the right-hand side so we subtract 2x first. That leaves us with 6x - 22 = 8. So, the next step is to add 22 to both sides. That leaves us with 6x = 30. Finally, we have to divide by 6 leaving x = 5. Did you get them right?
  • Question 4

There are three steps to solving the following equation

23 - 7y = 35 - 13y

The steps are given below. Can you put them in the correct order? 

CORRECT ANSWER

Column A

Column B

1st
Add 13x to both sides.
2nd
Subtract 23 from both sides.
3rd
Divide both sides by 6.
EDDIE SAYS
The smallest number of y's is -13y on the right-hand side. Since this is negative we add 13y to both sides first. This leaves us with 23 + 6y = 35. The next step is to subtract 23 from both sides leaving 6y = 12. Finally, we divide by 6 leaving y = 2. Right, now to try solving some equations yourself.
  • Question 5

Which of the following is the correct solution to the equation

2x + 3 = 5x - 2?

CORRECT ANSWER
x = 5/3
EDDIE SAYS
The smallest number of x's is on the left-hand side so we subtract 2x from both sides to start. Here's the working 2x + 3 = 5x - 2 -2x -2x 3 = 3x - 2 +2 +2 5 = 3x ÷5 ÷5 5/3 = x Now since 3 does not divide into 5 exactly this answer is best left as a top-heavy fraction. Do not convert it to a decimal with a calculator. In the exam, you get full marks for a top-heavy fraction. Halfway through now. How are you doing?
  • Question 6

When solving the equation 

20 - 7x = 6x - 6

which of the following is the correct solution?

 

CORRECT ANSWER
x = 2
EDDIE SAYS
Remember when there is a '-' in front of the x we add it on. Here's the full working for this one. 20 - 7x = 6x - 6 + 7x +7x 20 = 13x - 6 + 6 = + 6 26 = 13x ÷13 ÷13 2 = x Getting the hang of it? There's more to come!
  • Question 7

Match the following equations to their solutions.

 

CORRECT ANSWER

Column A

Column B

7a - 7 = 4a + 8
a = 5
8a - 24 = 12 - 4a
a = 3
6 - 4a = 10 - 5a
a = 4
3a + 6 = 4 - 4a
a = -2/7
EDDIE SAYS
There are 4 equations to solve here so, the working has been shortened to save space. 7a - 7 = 4a + 8 → -4a, then +7, then ÷3 → a = 5 8a - 24 = 12 - 4a → +4a, then +24, then ÷12 → a = 3 6 - 4a = 10 - 5a → +5a, then -6 → a = 4 3a + 6 = 4 - 4a → +4a, then -6, then ÷7 → a = -2/7 How's it going? Are you getting the hang of them yet?
  • Question 8

This equation has brackets on the right-hand side as well as x's on both sides. Expand (multiply out) the brackets, then solve as usual. Fill your solution in the space.

4(x - 2) = 3x + 5

CORRECT ANSWER
EDDIE SAYS
Remember, when expanding brackets you multiply everything in the brackets by the number in front. So, 4(x - 2) = 4x - 8. Then the rest of the working is as follows. 4x - 8 = 3x + 5 -3x -3x x - 8 = 5 +8 +8 x = 13 How was that one? Don't let the brackets put you off.
  • Question 9

Here's another equation with brackets, this time on both sides. Expand both sets of brackets, then proceed as usual. Fill in your solution below.

5(2 - p) = 2(4 - 2p)

CORRECT ANSWER
EDDIE SAYS
So this time we expand the brackets on both sides as follows. 5(2 - p) = 2(4 - 2p) 10 - 5p = 8 - 4p Now, since there are negative p's on both sides we add the smallest which is 5p. 10 - 5p = 8 - 4p +5p +5p 10 = 8 + p -8 -8 2 = p How was this one? Getting more difficult? Only one more to do!
  • Question 10

Solve the following equation by expanding the brackets first. Fill in your solution in the space leaving it as a fraction.

3(4y + 3) = 2(3y + 5) 

 

CORRECT ANSWER
EDDIE SAYS
This is a tricky one! First, expand the brackets. 3(4y + 3) = 2(3y + 5) 12y + 9 = 6y + 10 -6y -6y 6y + 9 = 10 -9 -9 6y = 1 ÷6 ÷6 y = 1/6 Did you remember to leave your answer as a fraction? Well done, you\'ve finished now!
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