 # Create and Solve Equations

In this worksheet, students will practise creating and solving an equation to solve a 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 you know how to use it correctly. One of its major advantages is that it allows you to work with things you don't know, be they lengths of sides, sizes of numbers or something entirely different.

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.

Example A rectangle has sides of (4x + 1) cm and 3 cm. If it has a perimeter of 16cm2. Find the area of the rectangle.

This question is a perfect of example of using an equation to solve a problem. The thing that gives it away is that we are asked for something and part of our question is an unknown (x).

Step 1: Draw a diagram if possible (or you aren't given one) Step 2: Create an expression for the thing you know.

In this question, we are given the perimeter. We know that perimeter is 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 I haven't made this into an equation. This is just because the question asks for an expression.

Step 3: Create an equation for the thing 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...

8x + 8 = 16

Step 4: Solve the equation:

We can just do this using our basic solving equations rules (do the same to both sides, do 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 found 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 x 3 = 15 cm2 For this rectangle, create an expression for the perimeter of the shape. This rectangle has a perimeter of 38. Use this to find an equation (in x) for the perimeter. 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. If all the bags have the same number of marbles in, create an expression for the total number of marbles James has.

Use m for the number of marbles in the bag

James has 4 bags of marbles and 5 loose marbles. If he has 41 marbles in total, create an equation for the total number of marbles he has.

Use m for the number of marbles in a bag

How many marbles were in each of the bags?

7

8

9

10

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Create an expression, in it's simplest form, for the sum of the angles

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Create an equation, in it's simplest form, for the sum of the angles

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Find the sizes of all of the angles

 Angle 2x + 14 4x + 8 5x - 7

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Angle 2x + 14 4x + 8 5x - 7
• Question 1 For this rectangle, create an expression for the perimeter of the shape.

8x + 6
8x+6
EDDIE SAYS
Perimeter means 'add all the sides together'. If we add together the algebra in the shape we get... 3x + 1 + x + 2 = 4x +3 Don't forget, however, that we have the same on the other two sides. Most people who lose marks here forget to do this part.
• Question 2 This rectangle has a perimeter of 38. Use this to find an equation (in x) for the perimeter.

8x+6=38
8x + 6 = 38
EDDIE SAYS
In the last question, we worked out that the perimeter was 8x + 6, all we have to do to create the equation is add =38 When you get a question like this, don't solve it yet. If the question asks for the equation, leave it as the equation.
• Question 3 Given that the perimeter of this shape is 38 cm. Find the lengths of each side.

EDDIE SAYS
We worked out that the equation for the perimeter is 8x + 6 = 38. If we solve this we find that x = 4. We can now just bang this into the expressions to find the length of each side.
• Question 4

James buys 4 bags of marbles and 5 loose marbles. If all the bags have the same number of marbles in, create an expression for the total number of marbles James has.

Use m for the number of marbles in the bag

4m+5
4m + 5
EDDIE SAYS
If we say that there are m marbles in a bag, there will be 4m marbles in the four bags. He also has 5 single marbles. Just put them together.
• Question 5

James has 4 bags of marbles and 5 loose marbles. If he has 41 marbles in total, create an equation for the total number of marbles he has.

Use m for the number of marbles in a bag

4x+5=41
4x + 5 = 41
EDDIE SAYS
We worked out that the expression for the number of marbles James has was 4m + 5. We've now been told that he has 41 in total. Once again, bang those values together.
• Question 6

How many marbles were in each of the bags?

9
EDDIE SAYS
We had the equation 4m + 5 = 41. We just now need to solve this equation.
• Question 7

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Create an expression, in it's simplest form, for the sum of the angles

11x+15
11x + 15
EDDIE SAYS
All we need to do here is add together these three terms. 4x - 8 + 5x -7 + 2x + 14
• Question 8

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Create an equation, in it's simplest form, for the sum of the angles

11x+15=180
11x + 15 = 180
EDDIE SAYS
We worked out before that the angles added up to 11x + 15. As this is a triangle, remember that the angles have to add up to 180°
• Question 9

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). Find the sizes of all of the angles

 Angle 2x + 14 4x + 8 5x - 7
EDDIE SAYS
If we solve the equation 11x + 15 = 180, we find that x = [email protected]; We can then just substitute 15° into the expressions to find the angles
• Question 10

A triangle has angles of (4x - 8), (5x -7) and (2x + 14). EDDIE SAYS
The correct definition of an isosceles triangle is that it has two sides the same. 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, two sides must also be, so we can use angles to define an isosceles triangle.
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