Solving Linear Equations

A number is hiding

Ama's mother sends her to the market with some money. Ama buys three oranges at GH₵2 each and comes back with GH₵4. How much did she take?

You can probably work this out in your head. But what if the problem were longer, with bigger numbers, or with two unknowns? You would want a method that always works.

That method is called solving an equation. The idea is simple: give the hidden number a name, write down what you know about it as a sentence with an equals sign, and then undo the operations one at a time until the hidden number stands alone.

In this book you will learn to:

• use a letter to stand for a number you do not know yet
• turn a sentence in words into an equation
• solve equations in one step and in two steps
• check your answer so you know it is right
• solve real problems about money, ages, lengths and shopping in Ghana

Let us start with the letter.

What is a variable?

A variable is a letter that stands for a number we do not know yet, or a number that can change.

If Kofi has some mangoes and we do not know how many, we can say he has m mangoes. If he gets 3 more, he has m + 3 mangoes. If he shares them equally with a friend, each gets m ÷ 2.

You can use any letter: x, y, n, t, a. Mathematicians love x, but a letter that reminds you of the thing is often easier: c for cost, a for age, d for distance.

Two rules to remember:

• 3m means 3 × m. We do not write the multiplication sign.
• m/4 or m ÷ 4 means m divided by 4.

Try this

Write an expression for each one:

1. Abena has b bananas and eats 2.
2. A pencil costs p cedis. What do 5 pencils cost?
3. Kwame's age a in 10 years' time.

Expressions and equations

An expression is a bit of maths with no equals sign: 3m + 2, x − 7, 5y.

An equation says that two things are equal. It always has an equals sign with something on each side:

3m + 2 = 14

The left-hand side is 3m + 2. The right-hand side is 14. The equation claims that these two are the same number. Our job is to find the value of m that makes the claim true.

A linear equation is one where the variable is not squared or multiplied by itself. x appears on its own, perhaps multiplied by a number or with numbers added or taken away. Everything in this book is linear.

Solving an equation means finding the value of the variable that makes both sides equal. For the equation above, m = 4, because 3 × 4 + 2 = 14. Later you will see how to get to that answer without guessing.

An equation is a balance

Think of a weighing scale with two pans, like the ones used to sell rice or tomatoes at the market. When the pans are level, the weight on the left equals the weight on the right.

An equation is the same. The equals sign is the pivot. Whatever you do to one side, you must do to the other, or the balance tips and the equation is no longer true.

If both pans hold 10 kg and you take 2 kg off the left, you must take 2 kg off the right to keep it level. If you double the left, you must double the right.

This is the golden rule of solving equations. Every step in this book obeys it. When you forget it, your answer goes wrong; when you follow it, you can solve any linear equation.

Say it aloud: Do the same to both sides.

Is it true? Substituting

Before we solve, let us learn to check. Checking means putting a number in place of the variable and seeing whether the two sides really are equal. This is called substituting.

Example. Is x = 5 a solution of 2x + 3 = 13?

Left side: 2 × 5 + 3 = 10 + 3 = 13. Right side: 13. Both sides are 13, so yes, x = 5 is a solution.

Example. Is y = 4 a solution of 3y − 1 = 14?

Left side: 3 × 4 − 1 = 12 − 1 = 11. Right side: 14. 11 is not 14, so y = 4 is not a solution.

Try this

Check whether each value is a solution:

1. a = 6 for a + 9 = 15
2. b = 7 for 4b = 24
3. c = 10 for c/2 + 1 = 6

Inverse operations

To find a hidden number we undo what has been done to it. Every operation has an opposite, called its inverse:

Operation	Inverse
add 5	subtract 5
subtract 5	add 5
multiply by 3	divide by 3
divide by 3	multiply by 3

Think of putting on socks and then shoes. To undo it, you take off the shoes first, then the socks. You reverse the order and reverse each action.

If a number was multiplied by 4 and then had 6 added, to get back to the number you first subtract 6, then divide by 4.

Try this

Write the inverse of each:

1. add 12
2. divide by 7
3. multiply by 9, then subtract 2 (two inverses, in the right order)

One-step equations: adding and subtracting

Example 1. Solve x + 7 = 12.

Something has been added to x. Undo it by subtracting 7 from both sides:

x + 7 − 7 = 12 − 7 x = 5

Check: 5 + 7 = 12. Correct.

Example 2. Solve y − 9 = 4.

Nine has been taken away. Undo it by adding 9 to both sides:

y − 9 + 9 = 4 + 9 y = 13

Check: 13 − 9 = 4. Correct.

Always write what you are doing to both sides, even if it feels slow. Clear working earns marks and stops mistakes.

Try this

1. n + 15 = 40
2. p − 6 = 21
3. 8 + k = 20

One-step equations: multiplying and dividing

Example 1. Solve 5m = 35.

Remember 5m means 5 × m. Undo the multiplying by dividing both sides by 5:

5m ÷ 5 = 35 ÷ 5 m = 7

Check: 5 × 7 = 35. Correct.

Example 2. Solve t/4 = 6.

The number has been divided by 4. Undo it by multiplying both sides by 4:

t/4 × 4 = 6 × 4 t = 24

Check: 24 ÷ 4 = 6. Correct.

Be careful with a common mistake: in 5m = 35, students sometimes subtract 5. But 5 is multiplying m, not being added, so you must divide.

Try this

1. 6x = 54
2. y/3 = 9
3. 12 = 4w
4. z/10 = 2.5

Recap: the basics

Let us pause and gather what we know.

• A variable is a letter standing for an unknown number.
• An equation has an equals sign; both sides are equal.
• To check, substitute the value and see if both sides match.
• An equation is a balance: do the same to both sides.
• Undo an operation with its inverse.

Equation	Undo by	Answer
x + 4 = 11	subtract 4	x = 7
x − 4 = 11	add 4	x = 15
4x = 20	divide by 4	x = 5
x/4 = 3	multiply by 4	x = 12

Quick quiz

Solve, and check each one:

1. a + 13 = 30
2. b − 11 = 2
3. 9c = 72
4. d/5 = 8

If you got all four, you are ready for two-step equations.

Two-step equations

Now the hidden number has had two things done to it. Take this:

3x + 4 = 19

Here x was multiplied by 3, and then 4 was added. We undo in reverse order: first undo the adding, then undo the multiplying.

Step 1. Subtract 4 from both sides.

3x + 4 − 4 = 19 − 4 3x = 15

Step 2. Divide both sides by 3.

3x ÷ 3 = 15 ÷ 3 x = 5

Check: 3 × 5 + 4 = 15 + 4 = 19. Correct.

The pattern is always: deal with the added or subtracted number first, then the multiplying or dividing number. Think of it as peeling an onion from the outside in.

Two-step with subtraction

Example. Solve 4y − 7 = 25.

Something was multiplied by 4 and then 7 was taken away.

Step 1. Add 7 to both sides.

4y − 7 + 7 = 25 + 7 4y = 32

Step 2. Divide both sides by 4.

y = 8

Check: 4 × 8 − 7 = 32 − 7 = 25. Correct.

Example. Solve 2n − 15 = 1.

2n = 16 (add 15 to both sides) n = 8 (divide both sides by 2)

Check: 2 × 8 − 15 = 16 − 15 = 1. Correct.

Try this

1. 5a − 3 = 22
2. 7b − 10 = 39
3. 3c − 8 = 10
4. 6d − 1 = 47

Two-step with division

Example. Solve x/3 + 5 = 9.

x was divided by 3 and then 5 was added. Undo the adding first.

Step 1. Subtract 5 from both sides.

x/3 = 4

Step 2. Multiply both sides by 3.

x = 12

Check: 12 ÷ 3 + 5 = 4 + 5 = 9. Correct.

Example. Solve m/5 − 2 = 6.

m/5 = 8 (add 2 to both sides) m = 40 (multiply both sides by 5)

Check: 40 ÷ 5 − 2 = 8 − 2 = 6. Correct.

A tip: when the variable is divided, students often forget the final multiplication and leave the answer as 8 or 4. Always ask, Is the variable alone yet?

Try this

1. x/2 + 3 = 10
2. y/4 − 1 = 5
3. k/6 + 7 = 9

Setting out your working

Teachers and examiners want to see how you got your answer. Here is a tidy layout that works every time.

Solve 6x − 9 = 27

6x − 9 + 9 = 27 + 9 (add 9 to both sides) 6x = 36 6x ÷ 6 = 36 ÷ 6 (divide both sides by 6) x = 6

Check: 6(6) − 9 = 36 − 9 = 27 ✓

Notice:

• Each line has one step.
• The equals signs line up underneath each other.
• What you did is written in brackets.
• The answer is clearly marked.
• A check comes at the end.

Writing 6(6) means 6 × 6; brackets are another way of showing multiplication.

Try this

Set out the full working, with a check, for 8y + 5 = 61.

Negative answers

Sometimes the hidden number is negative. Do not panic; the method is exactly the same.

Example. Solve x + 9 = 4.

x = 4 − 9 x = −5

Check: −5 + 9 = 4. Correct.

Example. Solve 3y + 10 = 1.

3y = 1 − 10 = −9 y = −9 ÷ 3 = −3

Check: 3 × (−3) + 10 = −9 + 10 = 1. Correct.

Remember the sign rules:

• A positive times a negative is negative.
• A negative times a negative is positive.
• Subtracting a larger number from a smaller one gives a negative.

Try this

1. a + 12 = 5
2. 2b + 8 = 2
3. c − 3 = −7

Fraction and decimal answers

The answer need not be a whole number. Treat fractions and decimals calmly.

Example. Solve 4x = 10.

x = 10 ÷ 4 = 2.5 (or 5/2)

Check: 4 × 2.5 = 10. Correct.

Example. Solve 2y + 1 = 8.

2y = 7 y = 3.5

Check: 2 × 3.5 + 1 = 7 + 1 = 8. Correct.

Example. Solve 3n − 2 = 2.

3n = 4 n = 4/3 (leave it as a fraction; 1.33 is only approximate)

When the division does not come out exactly, a fraction is the exact answer. Write 4/3, not 1.3, unless the question asks for a decimal to a given number of places.

Try this

1. 5a = 12
2. 2b − 3 = 4
3. 8c + 2 = 5

Brackets in equations

Sometimes an equation has brackets:

3(x + 2) = 21

This means 3 lots of (x + 2). There are two ways to solve it.

Way 1: divide first. Divide both sides by 3.

x + 2 = 7 x = 5

Way 2: expand first. Multiply out the bracket: 3 × x + 3 × 2 = 3x + 6.

3x + 6 = 21 3x = 15 x = 5

Both give x = 5. Check: 3(5 + 2) = 3 × 7 = 21. Correct.

Way 1 is quicker when the number outside divides neatly into the right-hand side. Way 2 always works.

Try this

1. 2(y + 5) = 18
2. 4(n − 1) = 20
3. 5(k + 3) = 40

Recap: two-step equations

You can now handle any equation of the forms ax + b = c, ax − b = c, x/a + b = c and a(x + b) = c.

The routine:

1. Look at what was done to the variable.
2. Undo the adding or subtracting first.
3. Then undo the multiplying or dividing.
4. Write every step, doing the same to both sides.
5. Check by substituting.

Mixed practice

1. 3x + 7 = 25
2. 5y − 4 = 31
3. m/4 + 6 = 11
4. 2(p − 3) = 14
5. 7q + 2 = 2
6. 9r − 5 = 4

Answers for you to check your checking: 6, 7, 20, 10, 0, 1.

If a question gave you trouble, go back to its page before moving on. The next part of the book is about words, and it builds on these skills.

From words to symbols

Most real problems arrive as sentences, not equations. Our job is to translate.

Words	Symbols
a number	x
5 more than a number	x + 5
5 less than a number	x − 5
twice a number	2x
three times a number	3x
half a number	x/2
a number increased by 8	x + 8
a number decreased by 8	x − 8
the sum of a number and 4	x + 4
the product of 6 and a number	6x

Watch the order carefully. "5 less than a number" is x − 5, not 5 − x. "Subtract 5 from a number" is also x − 5.

Try this

Write each as an expression:

1. 9 more than a number
2. a number multiplied by 7
3. 4 less than twice a number

Equals: the key word

An equation needs an equals sign, and in words the equals sign hides behind phrases like is, equals, makes, gives, the result is, the total is.

Example. "When 6 is added to a number, the result is 15."

• "a number" → x
• "6 is added" → x + 6
• "the result is 15" → = 15

x + 6 = 15, so x = 9.

Example. "Three times a number, less 4, is 20."

3x − 4 = 20 3x = 24 x = 8

Check: 3 × 8 − 4 = 24 − 4 = 20. Correct.

Try this

Form and solve an equation for each:

1. A number increased by 11 is 30.
2. Twice a number is 46.
3. Five times a number, plus 3, gives 38.
4. Half a number is 13.

The four-step plan

Every word problem can be tackled with the same plan. Learn it by heart.

1. Choose. Decide what the unknown is and give it a letter. Write a sentence: Let x be the number of …

2. Form. Translate the story into an equation.

3. Solve. Use the steps you know.

4. Answer and check. Go back to the story, write the answer in words with the correct units (cedis, years, metres), and check it makes sense.

Example. Abena thinks of a number, doubles it and adds 7. She gets 31. What was her number?

1. Let n be Abena's number.
2. 2n + 7 = 31
3. 2n = 24, so n = 12.
4. Abena's number was 12. Check: 2 × 12 + 7 = 31. ✓

The first step is the one students skip most often, and it is the one that stops confusion. Always say what your letter means.

Money problems in cedis

Example. Kofi buys 4 exercise books at the same price and pays GH₵18 in total. What does one book cost?

1. Let b be the cost of one book in cedis.
2. 4b = 18
3. b = 18 ÷ 4 = 4.5
4. One book costs GH₵4.50. Check: 4 × 4.50 = 18. ✓

Example. Ama has GH₵50. She buys a bag and has GH₵23 left. What did the bag cost?

1. Let c be the cost of the bag.
2. 50 − c = 23
3. c = 50 − 23 = 27
4. The bag cost GH₵27. Check: 50 − 27 = 23. ✓

Notice we wrote money as 4.50, not 4.5, in the final answer. Cedis are written with two decimal places for pesewas.

Try this

Kwame buys 6 pens and pays GH₵21. Find the cost of one pen.

Money problems: two steps

Example. A trotro fare is GH₵3 and Abena also pays GH₵5 for a snack. If she does the same trip for several days and spends GH₵29 on fares plus the one snack, how many trips did she make?

1. Let t be the number of trips.
2. 3t + 5 = 29
3. 3t = 24, so t = 8.
4. She made 8 trips. Check: 3 × 8 + 5 = 29. ✓

Example. Kofi sells oranges at GH₵2 each. He started the day with GH₵15 in his box and ended with GH₵47. How many oranges did he sell?

1. Let n be the number of oranges sold.
2. 15 + 2n = 47
3. 2n = 32, so n = 16.
4. He sold 16 oranges. Check: 15 + 2 × 16 = 47. ✓

Try this

A tailor charges GH₵40 to sew a dress plus GH₵12 for each extra pocket. Ama pays GH₵76. How many pockets did she ask for?

Age problems

Ages are a favourite in exams. The key: everyone ages at the same speed, so "in 5 years" means add 5 to every person's age.

Example. Kwame is 3 years older than his sister Abena. Their ages add up to 29. How old is each?

1. Let a be Abena's age. Then Kwame is a + 3.
2. a + (a + 3) = 29
3. 2a + 3 = 29, so 2a = 26, a = 13.
4. Abena is 13, Kwame is 16. Check: 13 + 16 = 29. ✓

Example. Ama's mother is 4 times as old as Ama. In 6 years' time the mother will be 40. How old is Ama now?

1. Let Ama's age be x. Mother's age is 4x.
2. 4x + 6 = 40
3. 4x = 34, x = 8.5.
4. Ama is 8 and a half. Check: 4 × 8.5 + 6 = 40. ✓

Try this

Kofi is 5 years younger than his brother. Their ages total 31. Find both ages.

Perimeter and length problems

A rectangle's perimeter is the distance around it: 2 × length + 2 × width.

Example. A school garden is a rectangle. Its length is 4 m more than its width, and its perimeter is 36 m. Find the width.

1. Let the width be w metres. Length is w + 4.
2. Perimeter: 2w + 2(w + 4) = 36
3. 2w + 2w + 8 = 36, so 4w + 8 = 36, 4w = 28, w = 7.
4. Width 7 m, length 11 m. Check: 2 × 7 + 2 × 11 = 14 + 22 = 36. ✓

Example. A rope is cut into two pieces. One piece is 2.5 m longer than the other. The rope was 12 m. How long is the short piece?

1. Short piece s; long piece s + 2.5.
2. s + s + 2.5 = 12
3. 2s = 9.5, s = 4.75.
4. Short piece 4.75 m. ✓

Try this

A rectangle's length is three times its width. Its perimeter is 48 cm. Find the width.

Collecting like terms first

Sometimes the variable appears more than once on one side. Collect like terms before solving.

Example. Solve 2x + 5 + 3x = 30.

5x + 5 = 30 (2x + 3x = 5x) 5x = 25 x = 5

Check: 2(5) + 5 + 3(5) = 10 + 5 + 15 = 30. ✓

Example. Solve 7y − 2y + 4 = 19.

5y + 4 = 19 5y = 15 y = 3

Like terms are terms with the same letter: 2x and 3x are like terms; 2x and 5 are not. Only numbers go with numbers, and only xs go with xs.

Try this

1. 4a + 3a = 42
2. 6b − b + 2 = 27
3. m + m + m − 4 = 11

Variable on both sides

What if the letter appears on both sides?

5x + 3 = 2x + 15

Gather all the x terms on one side. Subtract 2x from both sides (so the x term stays positive):

3x + 3 = 15 3x = 12 x = 4

Check: Left: 5(4) + 3 = 23. Right: 2(4) + 15 = 23. Equal. ✓

Example. Solve 7y − 4 = 3y + 20.

4y − 4 = 20 (subtract 3y) 4y = 24 y = 6

Check: 7(6) − 4 = 38; 3(6) + 20 = 38. ✓

The rule for choosing: move the smaller variable term to the other side, and the equation stays friendly.

Try this

1. 6a + 1 = 4a + 9
2. 9b − 5 = 5b + 11
3. 3c + 10 = 7c − 2

Recap: word problems

You have learnt to translate, solve and check. The plan:

1. Choose a letter and say what it stands for.
2. Form the equation from the words.
3. Solve it.
4. Answer in words with units, and check against the story.

Mixed practice

1. A number multiplied by 6 then reduced by 5 is 43. Find it.
2. Kofi bought 3 identical shirts and a GH₵15 cap for GH₵96. Cost of a shirt?
3. Ama is twice as old as Kwame. Their ages total 36. How old is Kwame?
4. A rectangle's length is 5 cm more than its width; perimeter 38 cm. Find the width.
5. Solve 4(x − 2) = 2x + 6.

Answers: 8; GH₵27; 12; 7 cm; 7.

If any one was hard, re-read the page that matches it. The last section of the book goes a little further and then looks at common mistakes.

Equations with fractions

A fraction like (x + 3)/2 means the whole of x + 3 divided by 2. Multiply both sides by the bottom number first.

Example. Solve (x + 3)/2 = 7.

x + 3 = 14 (multiply both sides by 2) x = 11

Check: (11 + 3)/2 = 14/2 = 7. ✓

Example. Solve (2y − 1)/3 = 5.

2y − 1 = 15 2y = 16 y = 8

Check: (16 − 1)/3 = 15/3 = 5. ✓

Treat the bar of the fraction like a bracket. Everything above the line belongs together.

Try this

1. (a + 5)/4 = 3
2. (3b − 2)/5 = 2
3. (c − 7)/6 = 1

Function machines

Another way to picture a two-step equation is a function machine. A number goes in, the machine does things to it, and a number comes out.

IN → × 3 → + 4 → OUT

If 5 goes in: 5 × 3 = 15, then 15 + 4 = 19. So 19 comes out.

Solving an equation is running the machine backwards. If 19 comes out, what went in? Reverse the machine, swapping each operation for its inverse and reading right to left:

OUT → − 4 → ÷ 3 → IN

19 − 4 = 15, then 15 ÷ 3 = 5. So 5 went in. That is exactly how we solved 3x + 4 = 19.

Drawing the machine helps when you are unsure which operation to undo first: the last thing the machine did is the first thing you undo.

Try this

Draw the machine for 2x − 7 = 11, then reverse it to find x.

Equations from tables

Kwame notes how much a taxi charges for different distances:

Distance (km)	1	2	3	4
Fare (GH₵)	7	9	11	13

Each extra kilometre adds GH₵2, and before any distance there is a starting charge. Going back one step from 1 km: 7 − 2 = 5. So the rule is

fare = 2d + 5

Check: d = 3 gives 2 × 3 + 5 = 11. ✓

Now the equation is a tool. If Kwame paid GH₵23, how far did he travel?

2d + 5 = 23 2d = 18 d = 9 km

A table that goes up by the same amount each time always hides a linear equation. Find the step (the number multiplying d) and the start (the number added), and you have the rule.

Try this

A photocopier shop charges GH₵1.50 per page plus GH₵2 for binding. Write the rule, then find how many pages cost GH₵32.

Equations from patterns

Equations also describe patterns. A pattern of chairs in a hall has 4 chairs in the first row, then adds 3 chairs each row. The number of chairs in row n is:

c = 3n + 1

(Check: row 1 gives 3 + 1 = 4. Row 2 gives 7. Row 3 gives 10.)

Question. Which row has 31 chairs?

3n + 1 = 31 3n = 30 n = 10

Row 10 has 31 chairs.

Question. A phone plan costs GH₵5 per month plus GH₵0.50 per minute of calls. If a month's bill is GH₵17, how many minutes were used?

5 + 0.5m = 17 0.5m = 12 m = 24 minutes

Many formulas in science, business and sport are linear equations in disguise. Once you can solve them, you can work backwards from any answer to its cause.

Checking: why it matters

Here is a student's work on 3x − 4 = 11:

3x = 11 − 4 = 7 x = 7/3

The student subtracted instead of adding. A check would have caught it: 3 × 7/3 − 4 = 7 − 4 = 3, not 11.

The correct working:

3x = 11 + 4 = 15 x = 5

Check: 3 × 5 − 4 = 11. ✓

Checking takes ten seconds and turns a wrong answer into a right one. In an exam it is the cheapest mark you will ever earn.

Checking a word problem has two parts: does the number satisfy the equation, and does it make sense in the story? If you find that someone is −3 years old or a book costs GH₵0, go back and look for the slip.

Common mistakes to avoid

Mistake 1: Doing something to one side only. From 2x + 3 = 9, writing 2x = 9. The 3 vanished without being subtracted from the right. Correct: 2x = 6.

Mistake 2: Undoing in the wrong order. From 2x + 3 = 9, dividing by 2 first to get x + 3 = 4.5. (This can work, but only if you divide every term: x + 1.5 = 4.5.) Safer: subtract first.

Mistake 3: Subtracting instead of dividing. From 4x = 20, writing x = 16. The 4 multiplies, so divide: x = 5.

Mistake 4: Sign slips with negatives. From x + 8 = 3, writing x = 5. Correct: x = 3 − 8 = −5.

Mistake 5: Forgetting what the letter means. Answering "7" when the question asked for the length in metres. Always finish with units and a sentence.

A check catches every one of these.

Puzzle page

Try these puzzles with a friend. Form an equation for each.

1. The number trick. Think of a number. Double it. Add 10. Halve the result. Subtract your original number. Whatever number you started with, the answer is 5. Can you use n to show why?

2. The market stall. Abena sold plantain at GH₵1.50 a finger. She paid GH₵8 for transport and still made a profit of GH₵34. How many fingers of plantain did she sell?

3. The football team. Kofi scored 4 more goals than Kwame this season. Together they scored 22. How many did Kwame score?

4. The mystery bag. A bag of 60 sweets is shared among three friends. Ama gets twice as many as Kofi; Abena gets 6 more than Kofi. How many does Kofi get?

Hints: in puzzle 4, let Kofi have k sweets and write Ama's and Abena's shares in terms of k.

Solutions to the puzzles

1. Start with n. Double: 2n. Add 10: 2n + 10. Halve: n + 5. Subtract n: 5. The n always cancels, so the answer is always 5.

2. Let f be the number of fingers. Money in: 1.5f. Profit = money in − transport.

1.5f − 8 = 34 1.5f = 42 f = 28

She sold 28 fingers.

3. Let Kwame score g. Kofi scored g + 4.

g + g + 4 = 22 2g = 18 g = 9

Kwame scored 9.

4. Kofi k, Ama 2k, Abena k + 6.

k + 2k + k + 6 = 60 4k = 54 k = 13.5

Half a sweet! Perhaps one was broken. Kofi gets 13.5 sweets, Ama 27, Abena 19.5. The check: 13.5 + 27 + 19.5 = 60. ✓

Real-life answers are not always whole numbers. Say so, rather than rounding silently.

Rearranging a formula

A formula is an equation with several letters. The skills you have learnt let you make any letter the subject.

The perimeter of a rectangle is P = 2l + 2w. Suppose you know P and l and want w. Treat the other letters like numbers:

P − 2l = 2w (subtract 2l from both sides) w = (P − 2l)/2 (divide both sides by 2)

Example. A school field has perimeter 300 m and length 100 m. Find its width.

w = (300 − 200)/2 = 50 m

Another: the cost of a school trip is C = 15n + 200, where n is the number of pupils. How many pupils can go for GH₵800?

15n = 600 n = 40 pupils

Making a letter the subject is just solving an equation where some of the "numbers" are letters. Same steps, same golden rule.

Try this

Make t the subject of d = st (distance = speed × time). Then find t when d = 120 km and s = 40 km/h.

Equations in a spreadsheet

Linear equations also live in computers. In a spreadsheet you might type:

=3*A1+4

If cell A1 holds the value of x, this cell shows 3x + 4. By typing different numbers in A1 you can try values quickly and see when the result hits 19. This is called trial and improvement.

Try A1 = 4: result 16. Too small. Try 6: result 22. Too big. Try 5: result 19. Found it!

Trial and improvement works, but it is slow and only finds exact answers when they are easy to guess. The algebra you have learnt gets to the answer directly, every time, even when the answer is 4/3 or −2.5.

Still, a spreadsheet is a fine way to check your algebra, or to explore what happens when you change the numbers in a story.

Where you will meet this again

Linear equations are not a topic you finish; they are a tool you keep.

• In science you will rearrange formulas: speed = distance ÷ time, so time = distance ÷ speed.
• In geometry you will solve for a missing angle: angles in a triangle add to 180°, so if two are 70° and 2x, then 70 + 2x + x = 180.
• In business you will find how many items must be sold to cover costs.
• In Basic 9 you will solve simultaneous equations (two unknowns, two equations) and inequalities (where the equals sign becomes < or >).

Everything there uses the same golden rule: do the same to both sides, undo in reverse order, check at the end.

Try this

Angles in a triangle are 50°, x and 2x. Form an equation and find x.

Test yourself

Work through all of these without looking back. Time yourself: 20 minutes is good.

Section A: solve

1. x + 17 = 31
2. 8y = 56
3. z/6 = 7
4. 5a − 3 = 32
5. b/4 + 9 = 12
6. 3(c + 2) = 27
7. 4d + 5 = 2d + 17
8. (e − 1)/5 = 3

Section B: form and solve

9. Seven more than three times a number is 34. Find the number.
10. Kofi buys 5 pencils and a GH₵6 ruler for GH₵13.50. Cost of a pencil?
11. Ama's father is 30 years older than Ama. Their ages total 50. How old is Ama?
12. A rectangle is 3 cm longer than it is wide. Its perimeter is 26 cm. Find its width.

Mark yourself with the next page.

Test answers

Section A

1. x = 14
2. y = 7
3. z = 42
4. a = 7
5. b = 12
6. c = 7
7. d = 6
8. e = 16

Section B

9. 3n + 7 = 34, n = 9
10. 5p + 6 = 13.5, p = 1.5, so GH₵1.50
11. a + (a + 30) = 50, a = 10
12. 2w + 2(w + 3) = 26, 4w + 6 = 26, w = 5 cm

Scoring: 11–12 correct: excellent. 8–10: good; revisit the pages for the ones you missed. Below 8: work through the recaps again and try the Try this tasks a second time.

Did you check each answer by substituting? If not, do it now. It is the habit that separates students who sometimes get it right from those who always do.

Glossary

Variable — a letter standing for a number that is unknown or can change.

Expression — maths with no equals sign, e.g. 3x + 2.

Equation — a statement that two expressions are equal, e.g. 3x + 2 = 14.

Linear equation — an equation where the variable is not squared or multiplied by itself.

Solve — find the value of the variable that makes the equation true.

Solution — that value.

Substitute — replace a letter with a number.

Inverse operation — the operation that undoes another: adding undoes subtracting; multiplying undoes dividing.

Like terms — terms with the same letter, which can be added together.

Coefficient — the number in front of a variable: in 5x, the coefficient is 5.

Perimeter — the distance around the edge of a shape.

Check — substitute your answer back to confirm both sides are equal.

Learn more

• Khan Academy: Solving equations — videos and practice questions with instant feedback.
• Maths is Fun: Solving equations — a clear, friendly walk-through with the balance picture.
• Maths is Fun: Algebra word questions — turning sentences into equations, with many examples.
• Corbettmaths contents — find "Solving equations" for videos, worksheets and exam-style questions.

Watch more

Think about it

1. Why must you do the same thing to both sides of an equation? What would go wrong if you did not?
2. Two-step equations are undone in the reverse order to how they were built. Can you think of something in daily life that you undo in reverse order?
3. A friend says checking is a waste of time because she is "always right". What would you say to her?
4. Make up a word problem about something in your town that leads to the equation 4x + 6 = 50.
5. Can an equation have no solution? Try to solve x + 3 = x + 5 and explain what happens.