Forming and Solving Equations

What you'll learn

How to turn information from shapes and word problems into algebra.
How to collect like terms to make expressions simpler.
How to form an equation from a fact, such as a perimeter or angle total.
How to solve the equation and answer the actual question asked.

1. The algebra basics you need

Before solving anything, you need to understand what the letters mean.

Definition

Key algebra words

A variable is a letter that stands for an unknown number, such as $x$ .
An expression is a calculation involving numbers and/or variables, such as $3 x + 5$ .
An equation says two expressions are equal, such as $3 x + 5 = 20$ .
Like terms are terms with the same variable part, such as $2 x$ and $5 x$ .

When you form equations, the first job is often to write an expression by adding or combining terms.

Collecting like terms

To collect like terms means to combine terms that match. For example, $3 x + 2 x = 5 x$ .

Example

Writing a perimeter expression

A triangle has side lengths $4 x - 2$ , $x + 6$ , and $2 x + 1$ cm. Write an expression for its perimeter.

Triangle with all three algebraic side lengths labelled, showing that the perimeter is the total around the outside.

Perimeter means the total distance around the outside, so add the three sides:
$P = (4 x - 2) + (x + 6) + (2 x + 1)$
Collect the $x$ terms: $4 x + x + 2 x = 7 x$ .
Collect the number terms: -2 + 6 + 1 = 5.
So the perimeter is $7 x + 5$ cm.

Common Mistake

Dropping a minus sign

In an expression like $3 x - 5$ , the minus belongs to the 5. When collecting number terms, treat it as negative 5.

2. Forming an equation

To form an equation, you use the information given. If you are told a perimeter, area, angle total, or total number of objects, set your expression equal to that value.

Key Idea

The main idea

Write an expression first, then use the given fact to make it equal to something.

Solving with inverse operations

Definition

Inverse operations

Inverse operations undo each other. Adding and subtracting are inverses. Multiplying and dividing are inverses.

You solve an equation by doing the same operation to both sides.

Example

Rectangle perimeter equation

A rectangle has length $3 x + 2$ cm and width $x + 4$ cm. Its perimeter is 44 cm. Find $x$ .

Rectangle with opposite sides equal, using two lengths and two widths to form the perimeter equation.

A rectangle has two lengths and two widths:
$P = 2 (3 x + 2) + 2 (x + 4)$
Expand the brackets, which means multiply everything inside:
$P = 6 x + 4 + 2 x + 8$
Collect like terms:
$P = 8 x + 12$
Use the given perimeter to form an equation:
$8 x + 12 = 44$
Subtract 12 from both sides, then divide by 8:
$8x+12=448x=32x=4\begin{aligned} 8x+12&=44\\ 8x&=32\\ x&=4 \end{aligned}$

Tip

Check your answer

Substitute your value of $x$ back into the original lengths to see if the perimeter works.

3. Equations from angle facts

Some questions give angles instead of lengths. You need to know the key angle facts.

Angles on a straight line add to 180°.
Angles around a point add to 360°.
A right angle is 90°.
Angles in a triangle add to 180°.

Example

A right angle split into two parts

A right angle is split into two angles labelled $4 x$ and $2 x + 6$ . Find $x$ .

A 90° angle split into two adjacent algebraic angles whose sum forms the equation.

A right angle is 90°, so the two angles add to 90:
$4 x + (2 x + 6) = 90$
Collect like terms:
$6 x + 6 = 90$
Subtract 6 from both sides:
$6 x = 84$
Divide by 6:
$x = 14$

Example

Angles in a triangle

The angles in a triangle are $2 x + 5$ , $x + 20$ , and $x - 1$ . Find $x$ .

Triangle with each interior angle labelled to show that the three angles add to 180°.

Angles in a triangle add to 180°:
$(2 x + 5) + (x + 20) + (x - 1) = 180$
Collect like terms:
$4 x + 24 = 180$
Subtract 24 from both sides:
$4 x = 156$
Divide by 4:
$x = 39$

Common Mistake

Forgetting the right angle

In a right-angled triangle, one angle is already 90°. The other two angles add to 90°, not 180° by themselves.

4. Equations from area and shape properties

Sometimes you must use a shape formula.

Rectangle area = length times width.
Triangle area = half times base times height.
Parallelogram area = base times perpendicular height.

Definition

Perpendicular height

The perpendicular height is the height measured at 90° to the base.

Perpendicular height is measured at right angles to the chosen base, not along a slanted side.

Example

Right-angled triangle area

A right-angled triangle has perpendicular sides $5 x$ cm and $4 x$ cm. Its area is 360 cm². Find $x$ .

Right-angled triangle where the two perpendicular sides are used as the base and height for the area formula.

Use the triangle area formula:
$area=12×base×height\text{area}=\frac{1}{2}\times \text{base}\times \text{height}$
Substitute the two perpendicular sides:
$12×5x×4x=360\frac{1}{2}\times 5x \times 4x=360$
Simplify the left-hand side:
$10x^2=360$
Divide by 10:
$x^2=36$
Take the positive square root:
$x = 6$

Common Mistake

Lengths are positive

When solving an equation such as $x^2=36$ , the mathematical answers are $x = 6$ and $x = - 6$ , but a length cannot be negative, so use the positive value.

Example

Using opposite sides of a parallelogram

A parallelogram has opposite sides labelled $4 x + 1$ cm and $2 x + 13$ cm. Its perpendicular height is 6 cm. Find its area.

Parallelogram with equal opposite sides and a perpendicular height, showing the base needed for the area.

Opposite sides of a parallelogram are equal:
$4 x + 1 = 2 x + 13$
Subtract $2 x$ from both sides:
$2 x + 1 = 13$
Subtract 1 from both sides:
$2 x = 12$
Divide by 2:
$x = 6$
Find the base by substituting into one side:
$4 x + 1 = 4 (6) + 1 = 25$
Area of a parallelogram is base times perpendicular height:
$25×6=15025\times 6=150$

5. Word problems: choose the variable carefully

In word problems, decide what $x$ represents first. Usually choose the smallest or simplest unknown.

Example

Counters total problem

Sam has some counters. Tia has twice as many as Sam. Ravi has 4 more than Tia. Altogether, they have 49 counters. How many counters does Ravi have?

Bar model showing Sam as one part, Tia as two equal parts, and Ravi as two equal parts plus 4 counters.

Let $x$ be the number of counters Sam has.
Tia has twice as many, so Tia has $2 x$ .
Ravi has 4 more than Tia, so Ravi has $2 x + 4$ .
Add all three expressions and set the total equal to 49:
$x + 2 x + (2 x + 4) = 49$
Collect like terms:
$5 x + 4 = 49$
Subtract 4, then divide by 5:
$5x=45x=9\begin{aligned} 5x&=45\\ x&=9 \end{aligned}$
Ravi has $2 x + 4$ , so substitute $x = 9$ :
$2 (9) + 4 = 22$

Tip

Answer the question asked

If the question asks for Ravi’s counters, do not stop at $x = 9$ . Here, $x$ was Sam’s counters, so you still need to find Ravi’s number.

6. Ratios from equations

A ratio compares amounts in order. For example, the ratio 2 : 3 means for every 2 of one amount, there are 3 of another.

Example

Ages and a ratio

Alex is one age. Ben is three times Alex’s age. Cara is 5 years younger than Ben. Their ages add to 100. Find the ratio Alex : Ben : Cara.

Let Alex’s age be $x$ .
Ben’s age is $3 x$ .
Cara’s age is $3 x - 5$ .
Add the ages and set them equal to 100:
$x + 3 x + (3 x - 5) = 100$
Collect like terms:
$7 x - 5 = 100$
Add 5, then divide by 7:
$7x=105x=15\begin{aligned} 7x&=105\\ x&=15 \end{aligned}$
The ages are Alex = 15, Ben = 45, Cara = 40, so the ratio is 15 : 45 : 40.
Simplify the ratio by dividing each part by 5: 3 : 9 : 8.

Exam technique

In the exam

Define your variable clearly, for example: let $x$ be the smallest angle.
Write an expression before trying to solve; most method marks come from the equation.
Check your final answer fits the original situation, especially if the question asks for an angle, length, area, or ratio.

Self review

Check yourself

Which angle fact would you use for the angles inside a triangle?
If a rectangle has sides $2 x + 1$ and $x + 4$ , what expression represents its perimeter?
In a word problem, how do you decide what $x$ should stand for?

Forming and Solving Equations

What you'll learn

How to turn information from shapes and word problems into algebra.
How to collect like terms to make expressions simpler.
How to form an equation from a fact, such as a perimeter or angle total.
How to solve the equation and answer the actual question asked.

1. The algebra basics you need

Before solving anything, you need to understand what the letters mean.

Definition

Key algebra words

A variable is a letter that stands for an unknown number, such as $x$ .
An expression is a calculation involving numbers and/or variables, such as $3 x + 5$ .
An equation says two expressions are equal, such as $3 x + 5 = 20$ .
Like terms are terms with the same variable part, such as $2 x$ and $5 x$ .

When you form equations, the first job is often to write an expression by adding or combining terms.

Collecting like terms

To collect like terms means to combine terms that match. For example, $3 x + 2 x = 5 x$ .

Example

Writing a perimeter expression

A triangle has side lengths $4 x - 2$ , $x + 6$ , and $2 x + 1$ cm. Write an expression for its perimeter.

Triangle with all three algebraic side lengths labelled, showing that the perimeter is the total around the outside.

Perimeter means the total distance around the outside, so add the three sides:
$P = (4 x - 2) + (x + 6) + (2 x + 1)$
Collect the $x$ terms: $4 x + x + 2 x = 7 x$ .
Collect the number terms: -2 + 6 + 1 = 5.
So the perimeter is $7 x + 5$ cm.

Common Mistake

Dropping a minus sign

In an expression like $3 x - 5$ , the minus belongs to the 5. When collecting number terms, treat it as negative 5.

2. Forming an equation

To form an equation, you use the information given. If you are told a perimeter, area, angle total, or total number of objects, set your expression equal to that value.

Key Idea

The main idea

Write an expression first, then use the given fact to make it equal to something.

Solving with inverse operations

Definition

Inverse operations

Inverse operations undo each other. Adding and subtracting are inverses. Multiplying and dividing are inverses.

You solve an equation by doing the same operation to both sides.

Example

Rectangle perimeter equation

A rectangle has length $3 x + 2$ cm and width $x + 4$ cm. Its perimeter is 44 cm. Find $x$ .

Rectangle with opposite sides equal, using two lengths and two widths to form the perimeter equation.

A rectangle has two lengths and two widths:
$P = 2 (3 x + 2) + 2 (x + 4)$
Expand the brackets, which means multiply everything inside:
$P = 6 x + 4 + 2 x + 8$
Collect like terms:
$P = 8 x + 12$
Use the given perimeter to form an equation:
$8 x + 12 = 44$
Subtract 12 from both sides, then divide by 8:
$8x+12=448x=32x=4\begin{aligned} 8x+12&=44\\ 8x&=32\\ x&=4 \end{aligned}$

Tip

Check your answer

Substitute your value of $x$ back into the original lengths to see if the perimeter works.

3. Equations from angle facts

Some questions give angles instead of lengths. You need to know the key angle facts.

Angles on a straight line add to 180°.
Angles around a point add to 360°.
A right angle is 90°.
Angles in a triangle add to 180°.

Example

A right angle split into two parts

A right angle is split into two angles labelled $4 x$ and $2 x + 6$ . Find $x$ .

A 90° angle split into two adjacent algebraic angles whose sum forms the equation.

A right angle is 90°, so the two angles add to 90:
$4 x + (2 x + 6) = 90$
Collect like terms:
$6 x + 6 = 90$
Subtract 6 from both sides:
$6 x = 84$
Divide by 6:
$x = 14$

Example

Angles in a triangle

The angles in a triangle are $2 x + 5$ , $x + 20$ , and $x - 1$ . Find $x$ .

Triangle with each interior angle labelled to show that the three angles add to 180°.

Angles in a triangle add to 180°:
$(2 x + 5) + (x + 20) + (x - 1) = 180$
Collect like terms:
$4 x + 24 = 180$
Subtract 24 from both sides:
$4 x = 156$
Divide by 4:
$x = 39$

Common Mistake

Forgetting the right angle

In a right-angled triangle, one angle is already 90°. The other two angles add to 90°, not 180° by themselves.

4. Equations from area and shape properties

Sometimes you must use a shape formula.

Rectangle area = length times width.
Triangle area = half times base times height.
Parallelogram area = base times perpendicular height.

Definition

Perpendicular height

The perpendicular height is the height measured at 90° to the base.

Perpendicular height is measured at right angles to the chosen base, not along a slanted side.

Example

Right-angled triangle area

A right-angled triangle has perpendicular sides $5 x$ cm and $4 x$ cm. Its area is 360 cm². Find $x$ .

Right-angled triangle where the two perpendicular sides are used as the base and height for the area formula.

Use the triangle area formula:
$area=12×base×height\text{area}=\frac{1}{2}\times \text{base}\times \text{height}$
Substitute the two perpendicular sides:
$12×5x×4x=360\frac{1}{2}\times 5x \times 4x=360$
Simplify the left-hand side:
$10x^2=360$
Divide by 10:
$x^2=36$
Take the positive square root:
$x = 6$

Common Mistake

Lengths are positive

When solving an equation such as $x^2=36$ , the mathematical answers are $x = 6$ and $x = - 6$ , but a length cannot be negative, so use the positive value.

Example

Using opposite sides of a parallelogram

A parallelogram has opposite sides labelled $4 x + 1$ cm and $2 x + 13$ cm. Its perpendicular height is 6 cm. Find its area.

Parallelogram with equal opposite sides and a perpendicular height, showing the base needed for the area.

Opposite sides of a parallelogram are equal:
$4 x + 1 = 2 x + 13$
Subtract $2 x$ from both sides:
$2 x + 1 = 13$
Subtract 1 from both sides:
$2 x = 12$
Divide by 2:
$x = 6$
Find the base by substituting into one side:
$4 x + 1 = 4 (6) + 1 = 25$
Area of a parallelogram is base times perpendicular height:
$25×6=15025\times 6=150$

5. Word problems: choose the variable carefully

In word problems, decide what $x$ represents first. Usually choose the smallest or simplest unknown.

Example

Counters total problem

Sam has some counters. Tia has twice as many as Sam. Ravi has 4 more than Tia. Altogether, they have 49 counters. How many counters does Ravi have?

Bar model showing Sam as one part, Tia as two equal parts, and Ravi as two equal parts plus 4 counters.

Let $x$ be the number of counters Sam has.
Tia has twice as many, so Tia has $2 x$ .
Ravi has 4 more than Tia, so Ravi has $2 x + 4$ .
Add all three expressions and set the total equal to 49:
$x + 2 x + (2 x + 4) = 49$
Collect like terms:
$5 x + 4 = 49$
Subtract 4, then divide by 5:
$5x=45x=9\begin{aligned} 5x&=45\\ x&=9 \end{aligned}$
Ravi has $2 x + 4$ , so substitute $x = 9$ :
$2 (9) + 4 = 22$

Tip

Answer the question asked

If the question asks for Ravi’s counters, do not stop at $x = 9$ . Here, $x$ was Sam’s counters, so you still need to find Ravi’s number.

6. Ratios from equations

A ratio compares amounts in order. For example, the ratio 2 : 3 means for every 2 of one amount, there are 3 of another.

Example

Ages and a ratio

Alex is one age. Ben is three times Alex’s age. Cara is 5 years younger than Ben. Their ages add to 100. Find the ratio Alex : Ben : Cara.

Let Alex’s age be $x$ .
Ben’s age is $3 x$ .
Cara’s age is $3 x - 5$ .
Add the ages and set them equal to 100:
$x + 3 x + (3 x - 5) = 100$
Collect like terms:
$7 x - 5 = 100$
Add 5, then divide by 7:
$7x=105x=15\begin{aligned} 7x&=105\\ x&=15 \end{aligned}$
The ages are Alex = 15, Ben = 45, Cara = 40, so the ratio is 15 : 45 : 40.
Simplify the ratio by dividing each part by 5: 3 : 9 : 8.

Exam technique

In the exam

Define your variable clearly, for example: let $x$ be the smallest angle.
Write an expression before trying to solve; most method marks come from the equation.
Check your final answer fits the original situation, especially if the question asks for an angle, length, area, or ratio.

Self review

Check yourself

Which angle fact would you use for the angles inside a triangle?
If a rectangle has sides $2 x + 1$ and $x + 4$ , what expression represents its perimeter?
In a word problem, how do you decide what $x$ should stand for?

Forming and Solving Equations

What you'll learn

1. The algebra basics you need

Collecting like terms

2. Forming an equation

Solving with inverse operations

3. Equations from angle facts

4. Equations from area and shape properties

5. Word problems: choose the variable carefully

6. Ratios from equations

Recap questions

You've reached the end

Forming and Solving Equations

What you'll learn

1. The algebra basics you need

Collecting like terms

2. Forming an equation

Solving with inverse operations

3. Equations from angle facts

4. Equations from area and shape properties

5. Word problems: choose the variable carefully

6. Ratios from equations

Recap questions

You've reached the end

Forming and Solving Equations

What you'll learn

1. The algebra basics you need

Collecting like terms

2. Forming an equation

Solving with inverse operations

3. Equations from angle facts

4. Equations from area and shape properties

5. Word problems: choose the variable carefully

6. Ratios from equations

Recap questions

You've reached the end

Forming and Solving Equations Revision Guide

Forming and Solving Equations

What you'll learn

1. The algebra basics you need

Collecting like terms

2. Forming an equation

Solving with inverse operations

3. Equations from angle facts

4. Equations from area and shape properties

5. Word problems: choose the variable carefully

6. Ratios from equations

Recap questions

You've reached the end

Forming and Solving Equations Revision Guide