Solving Equations

Topic Overview

When solving an equation, you are trying to find a solution which will make a particular mathematical equation true or correct.

For example:

x - 3 = 2

If you put 5 in place of x;

5 - 3 = 2

This solution is true, therefore x = 5 is a solution of this equation.

In the example above, there is only one solution for x. However, during your maths working, you will be asked to solve various equations which have multiple solutions.

For example:

(x-4)(x-3)= 0

If x is 4, the solution to the equation is:

(4-4)(4-3)=0

= 0 x 1= 0

When x is 3, the solution to the equation is:

(3-4)(3-3)=0

= -1 x 0 = 0

Both of these solutions make the results of the equation true, therefore the solutions to this equation are x = 4 or 3

Key Concepts

In your working, you will be required to solve various mathematical equations. but as a rule you will be required to:

• Recognise the different symbols within algebraic equations
• Manipulate and solve algebraic equations
• Distinguish the different roles played by letter symbols in algebra, using the correct notation

Listed below are a series of summaries and worked examples to help you solidify your knowledge about how to solve different mathematical equations.

Worked Examples

1-Using Inverses
The most effective methods with which to solve mathematical equations is by using inverses. This involves undoing the steps which the equation is performing in order to find its solution(s).

When using inverses, there are several rules which you must follow:

• adding and subtracting are the opposite of one another
• multiplying and dividing are the opposite of one other
• when using inverses you must perform the same method on both sides of the equation

Example
(a) - Solve the equation: x - 8 = 14

Solution
(a) - In order to get x on its own, you must add 8 to both sides of the equation:

x- 8 (+8) = 14 (+8)

By adding 8 to both sides, you can get x on its own and find its solution:

x (8 - 8 = 0) = 14+ 8

x = 14 + 8

Therefore, x = 22

Example
(b) - Solve the equation: 2x + 9 = 21 to find the value of x

Solution
(b)
Get 2x on its own by subtracting 9 from both sides:

2x+ 9(-9) = 21 (-9)

2x = 12

To work out the value of x, divide both sides by 2:

2x ÷ 2 = 12 ÷ 2

Therefore, x = 6

2-Solving equations which have unknown values on both sides of the equation
During your maths working, you will also be required to solve an equation which has unknown values on both sides of the equation. Once again, you can use inverses to solve these types of equations and find the solutions of the unknown values.

Example
(a) - Solve the equation 5x + 5 = x + 13, and find the value of x.

Solution
(a)
Firstly, you must move all of the x terms onto the same side of the equation. To do this, you will need to subtract x from both sides:

5x( - x) + 5 = x (-x) + 13

4x + 5 = 13

Now you need to get 4x on its own by subtracting 5 from both sides:

4x + 5 (-5) = 13 (-5)

4x = 8

Finally, to find the value of x, you must divide both sides by 4:

4x ÷ 4 = 8 ÷ 4

Therefore, x = 2

3-Solving equations which contain brackets
Some of the equations which you will be asked to solve may include brackets. In these cases, you need to multiply out the brackets before you can solve the equation.

Example
(a) - Solve the equation: 4(y + 3) = 28 to find the value of y

Solution
(a) Multiply out the brackets:

4(y+3) = 28

(4 x y) + (4 x 3) = 28

4y + 12 = 28

From here, you can use inverses to solve the equation:

4y +12 (-12) = 28 (-12)

4y = 16

4y ÷ 4 = 16 ÷ 4

Therefore, y = 4

4-Working out multiple solutions to an equation
You may also be asked to solve an equation which has multiple solutions.

Example
(a) - Solve the equation (x + 2)(2x - 4) = 0 to find the value(s) of x

Solution
(a) - Look at each bracket individually and use inverses to find the value(s) of x :

(x+2) = 0

x + 2 = 0

x + 2 (-2) = 0 (-2)

Therefore x = -2

(2x- 4) = 0

2x - 4 = 0

2x -4 (+4) = 0 (+4)

2x = 4

2x ÷ 2 = 4 ÷ 2

Therefore x = 2

Therefore, for the equation (x + 2)(2x -4) = 0 , x = 2 or -2

Tips

1. When using inverses, always remember that you must perform the same method on both sides of the equation
2. Remember that adding and subtracting are the opposite of one another
3. Remember multiplying and dividing are the opposite of one other
4. When faced with an equation which has multiple solutions, look at each bracket individually and then solve them to find each value

Note:

Solving equations is an easy to follow process once you understand the basic principles of using inverses. Always remember that if you perform an action to one side, you must perform the same action to the other side of the equation. Moreover, you should always check your solutions by placing the values you have calculated into the equation. If your values give you a true result, you know they are correct. If they do not, you need to go back and solve the equation again. In these cases, it is important that you write down all of your workings out. By doing so, you can easily discover where you have gone wrong and correct it. Ultimately, if you practice using inverses, you can quickly become skilled at solving a wide range of mathematical equations!