Create and Solve Equations

In this worksheet, students will practise interpreting a scenario then creating and solving an equation to solve this problem.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:

QUESTION 1 of 10

Algebra is a wonderful thing, if we know how to use it correctly.

One of its major advantages is that it allows us to work with numbers which we don't know; we call these variables

The technique we will discuss today is how to create and solve an equation to find one of these things that we don't know.

e.g. A rectangle has sides of (4x + 1) cm and 3 cm.

It also has a perimeter of 16cm2.

Find the area of this rectangle.

This question is a perfect example of using an equation to solve a problem.

The thing that gives it away is that we are asked for something which is currently an unknown (x).

Step 1: Draw a diagram to help you if possible (or if you aren't already given one):

Step 2: Create an expression for the elements you know.

In this question, we are given the perimeter.

We know that perimeter is calculated by adding the sides together so let's do that with the algebra:

(4x + 1) + 3 + (4x + 1) + 3

8x + 8

It's worth noting that we have not made this into an equation.

This is only because the question asks for an expression on this occasion.

Step 3: Create an equation for the elements you know.

We have just worked out an expression for the perimeter, and we know that the perimeter is equal to 16.

This gives us the equation then:

8x + 8 = 16

Step 4: Solve the equation.

We can just do this using our basic equation rules: do the same to both sides, and apply the opposite operation.

8x + 8 = 16

8x + 8 - 8 = 16 - 8

8x = 8

8x ÷ 8 = 16 ÷ 8

x = 1

In this question, we are asked to find the area.

To do this, we need to find the lengths of the sides and then multiply them together.

Remember that we just calculated that x = 1.

Side 1: 4x + 1    -->    4(1) + 1 = 5 cm

Side 2: 3 cm

Therefore, we can see that the area of this rectangle is 5 × 3 = 15 cm2

In this activity, we will create expressions and equations based on problems then solve them to find variables and apply this knowledge to calculate new information.

Consider this rectangle:

For this rectangle, create an expression for its perimeter.

Look at this rectangle again:

This rectangle has a perimeter of 38

Use this information to create an equation in terms of x which represents the perimeter.

Look at this rectangle one more time:

Given that the perimeter of this shape is 38 cm, find the lengths of each side.

James buys 4 bags of marbles and 5 loose marbles.

All the bags have the same number of marbles in them.

Create an expression for the total number of marbles James has.

Use m to represent the number of marbles in each bag.

James buys 4 bags of marbles and 5 loose marbles.

All the bags have the same number of marbles in them, and he has 41 marbles in total.

Create an equation which expresses the total number of marbles James has.

Use m to represent the number of marbles in each bag.

Using the information you have learnt in the last two questions, work out how many marbles James has in each bag.

7

8

9

10

A triangle has angles represented by the expressions below:

Create an expression for the sum of the angles in this triangle.

Ensure that your expression is written in its simplest form.

A triangle has angles represented by the expressions below:

Create an equation for the sum of the angles in this triangle.

Ensure that your equation is written in its simplest form.

A triangle has angles represented by the expressions below:

Find the size of all of the angles in this triangle.

You do not need to type the degrees sign to accompany each number on this occasion.

 Angle in ° 2x + 14 4x + 8 5x - 7

A triangle has angles represented by the expressions below:

 Angle in ° 2x + 14 4x + 8 5x - 7
• Question 1

Consider this rectangle:

For this rectangle, create an expression for its perimeter.

8x + 6
8x+6
8x+ 6
8x +6
EDDIE SAYS
To find the perimeter we need to add all the sides of this rectangle together. Remember that even though two sides are not annotated, these will have the same value as the sides we have been provided with as they are identical. This gives us: (3x + 1) + (x + 2) + (3x + 1) + (x + 2) = 3x + 1 + x + 2 + 3x + 1 + x + 2 = 8x + 6 The question here asks for an expression, which means that there will be no equals sign present. Did you spot that?
• Question 2

Look at this rectangle again:

This rectangle has a perimeter of 38

Use this information to create an equation in terms of x which represents the perimeter.

8x + 6 = 38
8x+6=38
8x+ 6 = 38
8x +6 = 38
8x + 6= 38
8x + 6 =38
EDDIE SAYS
In the last question, we worked out that the perimeter was represented by this expression: 8x + 6 So all we have to do to convert this into an equation is add '= 38' to the end of this. When you get a question like this, don't solve it yet. If the question asks for the equation alone, leave it as the equation. In the next question, we are going to move on to solve this equation and find the length of the two sides...
• Question 3

Look at this rectangle one more time:

Given that the perimeter of this shape is 38 cm, find the lengths of each side.

EDDIE SAYS
We worked out previously that the equation for the perimeter is: 8x + 6 = 38 We need to solve this equation to find the value of x, then put this into the two expressions related to the sides to find their numerical value. Let's start by solving 8x + 6 = 38: 8x = 38 - 6 8x = 32 x = 32 ÷ 8 x = 4 Now we can now just bang this value for x into the expressions we have been given to find the length of each side: 3x + 1 = (3 × 4) + 1 = 13 x + 2 = 4 + 2 = 6 It's as simple as that! How did you get on with this first series of questions? Let's try another now...
• Question 4

James buys 4 bags of marbles and 5 loose marbles.

All the bags have the same number of marbles in them.

Create an expression for the total number of marbles James has.

Use m to represent the number of marbles in each bag.

4m + 5
4m+5
4m+ 5
4m +5
EDDIE SAYS
If we say that there are m marbles in each bag, there will be 4m marbles in the four bags. James also has 5 single marbles. If we put these two pieces of information together, we reach our expression: 4m + 5
• Question 5

James buys 4 bags of marbles and 5 loose marbles.

All the bags have the same number of marbles in them, and he has 41 marbles in total.

Create an equation which expresses the total number of marbles James has.

Use m to represent the number of marbles in each bag.

4m + 5 = 41
4m+5=41
4m+ 5 = 41
4m +5 = 41
4m + 5= 41
4m + 5 =41
EDDIE SAYS
We worked out that the expression for the total number of marbles in the last question, it was: 4m + 5 We've now been told that he has 41 in total. So we just need to add '= 41' to the end to convert this expression into an equation. Simple!
• Question 6

Using the information you have learnt in the last two questions, work out how many marbles James has in each bag.

9
EDDIE SAYS
We have already worked out the equation which represents the information we have learnt about James and his marbles so far: 4m + 5 = 41 To find out how many marbles are in each bag, we need to solve this equation to find the value of m. 4m + 5 = 41 4m = 41 - 5 4m = 36 m = 36 ÷ 4 m = 9 Did you calculate that value correctly?
• Question 7

A triangle has angles represented by the expressions below:

Create an expression for the sum of the angles in this triangle.

Ensure that your expression is written in its simplest form.

11x + 15
11x+15
11x+ 15
11x +15
EDDIE SAYS
How do we find the sum of the angles? We just need to add all three values together: (4x + 8) + (2x + 14) + (5x - 7) = 4x + 8 + 2x + 14 + 5x - 7 = 11x + 15
• Question 8

A triangle has angles represented by the expressions below:

Create an equation for the sum of the angles in this triangle.

Ensure that your equation is written in its simplest form.

11x + 15 = 180
11x+15=180
11x+ 15 = 180
11x +15 = 180
11x + 15= 180
11x + 15 =180
EDDIE SAYS
We worked out in the previous question, that the angles added up to: 11x + 15 But we haven't been told any new information to enable us to turn this expression into an equation, have we? We already know that the total value of all three internal angles in a triangle is 180°. So, in order to convert this expression into an equation, we just need to add '= 180' to the end of it like this: 11x + 15 = 180
• Question 9

A triangle has angles represented by the expressions below:

Find the size of all of the angles in this triangle.

You do not need to type the degrees sign to accompany each number on this occasion.

 Angle in ° 2x + 14 4x + 8 5x - 7
EDDIE SAYS
Here we need to solve our equation to find the value of x, then substitute this number into each expression to work out the value of each angle. If we solve the equation 11x + 15 = 180, we find that x = 15°. We can then substitute 15° into each expression to find the angles like this: 2x + 14 = (2 × 15) + 14 = 44° 4x + 8 = (4 × 15) + 8 = 68° 5x - 7 = (5 × 15) - 7 = 68° We can double-check these answers by checking that the total of all three is 180: 44 + 68 + 68 = 180 So we can be extra-sure that these answers are correct!
• Question 10

A triangle has angles represented by the expressions below:

EDDIE SAYS
The correct definition of an isosceles triangle is that it has two sides which are exactly the same length. We need to realise that we don't know what the sides are so we can't use this definition. But, if two angles are the same then two sides must also be the same, so we can use angles to define this as an isosceles triangle. You've reached the end of this activity - well done! Why not spend some more time practising creating expressions and equations in your everyday life? It's easy to do - just give it a try!
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