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Explainer Video

Tutor: Meera

Summary

Solving equations

​​In a nutshell

​An equation can be solved by rearranging. You usually want to find the value of the unknown variable, e.g. xxx. which makes the equation true. Think of an equation as a balancing scale, whatever you do to one side of the equation, you must do the same to the other.



Balancing scales

The equation

​x+5=13x+5=13x+5=13

can be represented by:

Maths; Algebra; KS4 Year 10; Solving equations


To solve the equation for xxx, you need to rearrange the equation so that xxx is by itself on one side of the equation, and all other numbers or terms should be on the other side. To get xxx by itself, +5+5+5 needs to be moved to the other side. To do this, think of the inverse operation, or the opposite of +5+5+5. The opposite of +5+5+5 is −5-5−5 so subtract 555 from both sides of the equation. This gives​


​x=13−5x=8‾\begin {aligned} x &= 13 -5 \\ x &=\underline{8} \end {aligned}xx​=13−5=8​​​​

This can be represented by:

Maths; Algebra; KS4 Year 10; Solving equations



Solving algebraically

​When solving an equation, think about what to do to both sides of the equation to get the unknown variable by itself.


Example 1

x+3=7−3−3x=4‾\begin {aligned}\qquad x + 3 &= 7 \\-3 \qquad & \qquad -3\end {aligned}\\\quad \quad\underline {x=4}x+3−3​=7−3​x=4​​​


When there is more than one number to move to the other side of the equation, it is important to move them in the right order. Use reverse BIDMAS to help.


Example 2

Solve the equation 

​3x+4=103x+4=103x+4=10


Here, the 333 with the xxx and +4+4+4 need to be moved to the other side. To decide which number to move across first, think about substituting a number in for xxx. You would take the value of xxx, multiply by 333 first, then +4+4+4 to the result. So when solving an equation, reverse the process. So move +4+4+4 to the other side first, then move 333 to the other side.


The opposite of +4+4+4 is −4-4−4, so subtract 444 from both sides. Then 333 is multiplying xxx, so do the inverse and divide by 333 to both sides.


3x+4=10−4−43x=6÷3÷3x=2‾\begin {aligned}\qquad 3x+4 &=10 \\-4 \qquad & \qquad -4 \\3x &= 6 \\\div3 \qquad & \qquad \div 3 \end {aligned}\\\quad \quad\underline {x=2}3x+4−43x÷3​=10−4=6÷3​x=2​​


Example 3

Solve 

​3x+32=11x−483x+32 = 11x-483x+32=11x−48​​


When there are multiple terms with xxx in the equation, on both sides, first move the xxx terms on one side, and the numbers onto the other side. Move the xxx's onto the side where there are more of them.​


3x+32=11x−48−3x−3x32=8x−48+48+4880=8x÷8÷810=xx=10‾\begin {aligned}\quad 3x+32 &= 11x -48 \\-3x \qquad & \qquad -3x \\32 &= 8x - 48 \\+48 \qquad & \qquad +48 \\80 &= 8x \\\div 8 \qquad & \qquad \div 8 \\10 &= x \end {aligned}\\\quad \underline {x=10}3x+32−3x32+4880÷810​=11x−48−3x=8x−48+48=8x÷8=x​x=10​​​


​

Inverse operations

The table of inverse operations helps work out what operation to do to both sides of the equation. For example, addition (+++) is the inverse of subtraction (−-−), multiplication (×\times×) is the inverse of division (÷\div÷) and squaring (x2x^2x2) is the inverse of square rooting (x\sqrt xx​).



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FAQs - Frequently Asked Questions

What does solving an equation mean?

Solving an equation means to find the value of x. It means you have to rearrange your equation to make x the subject, then it is possible to calculate its value.

What are inverse operations?

Inverse operations are operations that perform the opposite function. For example, addition and subtraction are inverse operations. Multiplication and division are inverse operations.

How can you solve an equation?

An equation can be solved by rearranging. We can think of an equation like a balancing scale. Whatever we do to one side of the equation, we must do the same to the other.

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