Solving simple equations
In an equation, letters stand for a missing number. To solve an equation, find the values of missing numbers. A typical exam question is:
Solve the equation 2a + 3 = 7
This means we need to find the value of a. The answer is a = 2
There are two methods you can use when solving this type of problem:
- Trial and improvement
- Using inverses
Trial and improvement
This method involves trying different values until we find one that works.
Look at the equation 2a + 3 = 7
To solve it:
Write down the equation: 2a +3 = 7
Then, choose a value for 'a' that looks about right and work out the equation. Try '3'.
a = 3, so 2 ×3 + 3 = 9.
Using '3' to represent 'a' makes the calculation more than 7, so choose a smaller number for 'a'.
Try a = 2
2 × 2 + 3 = 7
Which gives the right answer. So a = 2
Be systematic in your approach:
- choose a number
- work it out
- then move the number up or down
However, sometimes the answers are negatives or decimal fractions, and the trial and improvement method will take a long time. Luckily, there is a better method.
The best way to solve an equation is by using 'inverses', or undoing what the equation is doing.
To use this method to solve equations remember that:
- Adding and subtracting are the inverse (or opposite) of each other.
- Multiplying and dividing are the inverse of each other.
This method is explained in the following pages. But for now, here is how to solve the question in the above example using inverses:
- First, write down the expression:
- 2a + 3 = 7
- Then, undo the + 3 by subtracting 3. Remember, you need to do it to BOTH sides!
- 2a + 3 - 3 = 7 - 3,
- so 2a = 4.
- Undo the multiply by 2 by dividing by 2, again on both sides:
- 2a ÷ 2 = 4 ÷ 2
The answer is: a = 2
Getting the unknown on its own
Sometimes an equation will have multiples of an unknown - eg 3x = 12, 5b = 20 or 16y = 4
To solve these, you need to get the unknown on its own. Do this by dividing both sides.
Sometimes you will be asked to solve an equation with unknowns on both sides of the equation.
Remember that whatever you do to one side you must also do to the other.
Equations with brackets
If an equation has brackets in it one method of solving it is to multiply out the brackets first. For example:
Solve the equation 3(b + 2) = 15.
- Write down the equation:
- 3(b + 2) = 15
- Multiply out the brackets. Remember, everything inside the brackets gets multiplied by 3
- 3 × b + 3 ×2 = 15
- When you have multiplied out the brackets you get:
- 3b + 6 = 15
- Next, undo the + 6. In other words, do the inverse and subtract 6 from both sides.
- 3b + 6 - 6 = 15 - 6
- So 3b = 9
Therefore, to find out what b is you need to do the inverse of multiplying by 3 which is dividing by 3.
So b = 3
Factorising expressions is the reverse of multiplying out brackets - you try to put brackets into an expression. You should make sure you can do well in the test bite for multiplying out brackets before you work on this bite.
Factors are whole numbers or terms that divide into another number or term with no remainder.
- 1, 2, 3, 4, 6 and 12 are factors of 12
- 1, 2, 4, x, 2x and 4x are factors of 4x
Factor pairs are numbers or terms that multiply together to get a certain number or terms.
For example, the factor pairs of 12 are:
- 1 & 12
- 2 & 6
- 3 & 4
You might be asked to factorise a quadratic expression, in order to solve a quadratic equation
Factorising simple quadratic expressions
You can factorise a simple quadratic expression:
- where all the numbers are positive.
- where the coefficient of the first term (the x2) is 1 - i.e. where the number in front of the x2 is 1.
To find the numbers that go in each bracket, you look for a pair of numbers which multiply to give the last number and add together to give the middle number.
Factorise the expression
You need to find the numbers that go into each bracket.
Look for a pair of numbers which multiply to give the last number and add to give the middle number.
For this expression, we're looking for two numbers which multiply to give 12 and add to give 7. They are 3 and 4.
So the answer is: