Equations

Equations can often be solved by guessing.

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"I think of a number and add 6 to it."

"The result is 10. "

"What is the number?"

Most people can easily work out that the number is 4.

In mathematics, we would write an equation x + 6 = 10, where x stands for the number. We would then solve the equation to find the value of the number (x).

This sounds a long way of doing things but many equations are more difficult to solve than the one above.

There are several types of equations and several methods of solving them.

When solving equations each step should be written on a new line, and the equals signs should be kept directly underneath each other.

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  • These equations usually have one variable, like the problem above and one answer.
  • Equations can sometimes be solved by working out the answer in your head.
    This method usually works for simple equations only.
  • The aim when solving an equation is to get one of the variables on its own, on one side of the equation.
  • To remove a term from one side of an equation, carry out the opposite operation to bothsides of the equation.

    Remember: Adding and subtracting are opposite operations.

    Multiplying and dividing are opposite operations.

 

One Step Equations
Examples
Answers

x + 7 = 15

x + 7 = 15

x + 7 − 7 = 15 − 7               (subtract 7 from both sides)

x = 8

g − 3 = 12

g − 3 = 12

g − 3 + 3 = 12 + 3               (add 3 to both sides)

g = 15

3w = 15

3w = 15

3w⁄3 = 15⁄3    (divide both sides by 3)

w = 5

k⁄4 = 5 

k⁄4 = 5 

k⁄4 × 4 = 5 × 4   (multiply both sides by 4)

k = 20

 

Two Step Equations

 
Examples
Answers
  

3p + 10 = 22

3p + 10 = 22

3p + 10 − 10 = 22 − 10

3p = 12

p = 4

 

 

(subtract 10 from both sides)

(divide both sides by 3)

2g − 7 = 13

2g − 7 = 13

2g − 7 + 7 = 13 + 7

2g = 20

g = 10

 

(add 7 to both sides)

(divide both sides by 2)

 

The answer should then be checked by substituting it back into the equation to make sure that both sides have the same value. This step can be done mentally. e.g. For 3w = 15. The solution is w = 5. Check 3 × 5 = 15.

Equations with Brackets

Generally, it is better to expand any brackets first.

e.g.
3(x + 7) = 24
  
 
3x + 21 = 24
 (expand brackets)
 
3x + 21 − 21 = 24 − 21      
 (subtract 21 from both sides)
 
3x = 3
 (divide both sides by 3)
 
x = 1
  

Problem Solving

Many problems can be solved by making equations and solving them.

The general way to approach these problems is:

  • Assign a variable or letter to the unknown quantity in the problem.
  • Make up an equation from the information in the problem.
  • Solve the equation.

 

Example
Answer

Hemi thinks of a number, doubles it and subtracts 8.

The result is 32.

What is his number?

 

Let the unknown number be x.

The equation is:

2x − 8 = 32

Solving the equation:

2x − 8 = 32

            2x = 32 + 8

      2x = 40

        x = 20

Hemi's number is 20