Forming and solving equations is about turning words into maths and then finding the value of an unknown number.
There are two main skills you need to learn:
- Forming an equation — turning information in a question into an equation.
- Solving an equation — finding the value of the unknown, usually $x$.
Let’s work through both step by step.
Forming equations
The key is to turn words into mathematical expressions.
Useful words to remember
Add
5 more than $x$
$x+5$
Subtract
3 less than $x$
$x-3$
Multiply
4 times $x$
$4x$
Divide
$x$ divided by 2
$\frac{x}{2}$
Example of “Form an equation”
A number plus 7 equals 19. Form an equation.
Step 1: Let the unknown number be $x$.
Step 2: “Plus 7” means $x+7$.
Step 3: “Equals 19” means we put $=19$.
The equation is:
$\boxed{x+7=19}$
Example of “A number multiplied by 3”
Three times a number is 24. Form an equation.
Let the number be $x$.
Three times the number is $3x$.
Therefore:
\[ \boxed{3x=24} \]
Example: A rectangle
A rectangle has a length of $x+3$ cm and a width of $x$ cm. Its perimeter is 18cm. Form an equation.
Remember:
\[ \text{Perimeter}=2\times\text{length}+2\times\text{width} \]
Substitute the expressions:
\[ 2(x+3)+2x=18 \]
This is the equation.
Exam tip: You do not always need to solve the equation if the question only asks you to form it.
Solving equations
To solve an equation, you need to get \(x\) on its own.
The golden rule is:
Whatever you do to one side of the equation, you must do to the other side.
Example:
$x+5=12$
We want $x$ on its own.
Subtract 5 from both sides:
\[ x+5-5=12-5 \]
Therefore:
\[ \boxed{x=7} \]
Check: $7+5=12$, so the answer is correct.
Example:
$x-4=10$
Add 4 to both sides:
\[ x-4+4=10+4 \]
Therefore:
\[ \boxed{x=14} \]
Example:
$3x=21$
Here, $3x$ means $3\times x$.
Divide both sides by 3:
\[ \frac{3x}{3}=\frac{21}{3} \]
Therefore:
\[ \boxed{x=7} \]
Example:
\[\frac{x}{4}=5\]
Multiply both sides by 4:
\[ \frac{x}{4}\times4=5\times4 \]
Therefore:
\[ \boxed{x=20} \]
Equations with $x$ on both sides
These are also common in GCSE Grade 4 questions.
Example:
$3x+4=x+12$
Step 1: Subtract \(x\) from both sides.
\[ 2x+4=12 \]
Step 2: Subtract 4 from both sides.
\[ 2x=8 \]
Step 3: Divide both sides by 2.
\[ \boxed{x=4} \]
Check:
Left side: $3(4)+4=16$
Right side: $4+12=16$
Both sides are equal, so $x=4$ is correct.
Forming and solving equations from word problems
These questions require you to form an equation first and then solve it.
Example: An age problem
A number is increased by 6. The answer is 17. Find the number.
Step 1: Form the equation
Let the number be $x$.
\[ x+6=17 \]
Step 2: Solve the equation
Subtract 6 from both sides:
\[ x=17-6 \]
\[ \boxed{x=11} \]
Example: A money problem
A cinema ticket costs £$x$. Two tickets cost £18 altogether. Find the cost of one ticket.
Let the cost of one ticket be £$x$.
Form the equation:
\[ 2x=18 \]
Divide both sides by 2:
\[ \boxed{x=9} \]
Each ticket costs £9.
What to remember
- Forming: Translate the words into an equation.
- Adding: Subtract to undo the addition.
- Subtracting: Add to undo the subtraction.
- Multiplying: Divide to undo the multiplication.
- Dividing: Multiply to undo the division.
- Word problems: Define $x$, form the equation, solve it, and include units where needed.
The best way to improve is to practise simple equations first, then move on to word problems and equations with $x$ on both sides.