Home

Maths

Algebra

Solving equations

Solving equations

Videos

Summary

Exercises

Select Lesson

Exam Board

Select an option

Statistics


Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability


Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry


Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry


Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area


Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change


Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs


Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra


Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Double brackets: Expanding and factorising

Double and triple brackets - Higher

Solving equations

Expressions, equations, formulae, functions and identities

Writing formulae and equations from word problems

Writing formulae and equations from diagrams

Rearranging formulae

Factorising quadratics

The quadratic formula - Higher

Complete the square - Higher

Algebraic fractions - Higher

Sequences

Finding the nth term

Solving inequalities

Inequalities on graphs - Higher

Iteration - Higher

Simultaneous equations: elimination and substitution

Non-linear simultaneous equations - Higher

Algebraic proof - Higher

Composite and inverse functions - Higher

Number


Types of numbers

Order of operations: BODMAS

Multiplying and dividing by powers of 10

Multiplying and dividing whole numbers

Multiplying and dividing decimals

Negative numbers: add, subtract, multiply, divide

Prime numbers and prime factorisation

Multiples, factors and prime factors

LCM and HCF

Fractions

Fractions, decimals and percentages

Writing recurring decimals as fractions

Rounding: Integers, decimal places, significant figures

Estimation

Error intervals

Upper and lower bounds - Higher

Powers and roots: Square and cube numbers

Laws of indices: multiply, divide, brackets

Index laws: negative and fractional indices - Higher

Surds: Simplify, add and subtract - Higher

Rationalising surds - Higher

Standard form calculations

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​).



Read more

Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Inverse operations to check calculations

Inverse operations to check calculations

Solving equations

Solving equations

Jump Ahead

Score 80% to jump directly to the final unit.

This is the current lesson and goal (target) of the path

Solving equations

Solving equations

Final Test

Test reviewing all units to claim a reward planet.

Create an account to complete the exercises

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.

Beta

© 2020 - 2025 evulpo AG