Equations with brackets:Equations with bracketsIf 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
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 Trial and improvementYou might need to use this method if you are asked to solve an equation where there is no exact answer. You may also be asked to give the solution to a given number of decimal places or significant figures. The question should indicate the degree of accuracy required.
Find the answer to the equation x3 – 2x = 25 to one decimal place.
Solution
We are looking for a number to replace x, that when applied to the equation will give us 25. Start by guessing what x could be, then refine your answer based on your result. Set it out like this.
First we'll try: x = 3
33 - (2 x 3)
3 x 3 x 3 - (2 x 3)
27 - 6
= 21; too small
Second try: x = 4
43 - (2 x 4)
4 x 4 x 4 - (2 x 4)
64 - 8
= 56; much too high
We could use a number half way between 3 and 4, but our first tries suggest it will be closer to 3.
Third try: x = 3.2
3.23 - (2 x 3.2)
3.2 x 3.2 x 3.2 - (2 x 3.2)
32.768 - 6.4
= 26.368; too high
Fourth try: x = 3.15
3.153 - (2 x 3.15)
3.15 x 3.15 x 3.15 - (2 x 3.15)
31.255… - 6.3
= 24.955…; close
This means the actual value of x is greater than 3.15 but less than 3.2.
Since we've been told to give the answer correct to 1 decimal place, the answer we are looking for is 3.2. Simultaneous equationsSimultaneous equations are two equations with two unknowns. They are called simultaneous because they must both be solved at the same time.
The first step is to try to eliminate one of the unknowns.
ExampleSolve these simultaneous equations and find the values of x and y.