Linear equations - Year 7 Mathematics

Year 7 core

By the end of this topic you should be able to:

solve one-step linear equations using inverse operations,
solve two-step linear equations by undoing operations in reverse order,
verify a solution by substituting it back in,
translate a simple worded problem into an equation and solve it.

Worked example 0 Real-world example: saving for a goal

You want to buy a $500 gaming console. You already have $80 saved and add $25 each week. How many weeks until you can buy it?

Write the equation: $80 + 25w = 500$ .
Subtract $80$ : $25w = 420$ .
Divide by $25$ : $w = 16.8$ .
You can’t save a fraction of a week, so you need $17$ weeks.
Check: $80 + 25 \times 17 = 80 + 425 = 505$ . ✓ (enough with $5 spare).

Key idea: the equation translates the English sentence into maths; solving it means undoing the operations to find the unknown.

1. The balance idea

An equation is a balanced pair of scales: whatever is on the left equals whatever is on the right. If you change one side, you must do the exact same thing to the other side to keep the scales level.

x + 5 = 12 \quad\Longrightarrow\quad x + 5 - 5 = 12 - 5 \quad\Longrightarrow\quad x = 7.

2. Inverse operations

To “undo” an operation, apply its inverse.

Inverse operations

Addition and subtraction are inverses

$x + a = b \;\Rightarrow\; x = b - a$ . Example: $x + 7 = 12 \Rightarrow x = 5$ .

Multiplication and division are inverses

$ax = b \;\Rightarrow\; x = \dfrac{b}{a}$ . Example: $3x = 18 \Rightarrow x = 6$ .

3. One-step equations

Worked example 1 One-step addition/subtraction

Solve $\;x + 9 = 23$ .

Subtract $9$ from both sides: $x = 23 - 9 = 14$ .

Check: $14 + 9 = 23$ .

Worked example 2 One-step multiplication/division

Solve $\;5x = 45$ .

Divide both sides by $5$ : $x = 45 \div 5 = 9$ .

Check: $5 \times 9 = 45$ .

4. Two-step equations

When there are two operations, undo them in reverse order: deal with addition/subtraction first, then with multiplication/division.

Worked example 3 Two-step equation

Solve $\;3x + 4 = 19$ .

Subtract $4$ from both sides: $3x = 15$ .
Divide both sides by $3$ : $x = 5$ .

Check: $3 \times 5 + 4 = 15 + 4 = 19$ .

Worked example 4 Variable in a fraction

Solve $\;\dfrac{x}{4} + 3 = 8$ .

Subtract $3$ from both sides: $\dfrac{x}{4} = 5$ .
Multiply both sides by $4$ : $x = 20$ .

Check: $20 \div 4 + 3 = 5 + 3 = 8$ .

5. From words to equations

Identify the unknown, give it a letter, write down what the sentence says.

Worked example 5 A number puzzle

“I think of a number, double it, add $5$ , and the result is $23$ . What is the number?”

Let the number be $n$ .
Write the equation: $2n + 5 = 23$ .
Solve: $2n = 18$ , so $n = 9$ .

Check: $2 \times 9 + 5 = 23$ .

Practice: Year 7 core

All answers are natural numbers (positive whole numbers).

Fluency

One-step equations

Solve $\;x + 8 = 20$ .
Solve $\;y - 6 = 13$ .
Solve $\;4x = 28$ .
Solve $\;\dfrac{y}{3} = 7$ .
Solve $\;x + 17 = 30$ .
Solve $\;a - 9 = 1$ .
Solve $\;7m = 42$ .
Solve $\;\dfrac{x}{6} = 4$ .

Fluency

Two-step equations

Solve $\;2x + 3 = 11$ .
Solve $\;3y - 5 = 7$ .
Solve $\;5x + 1 = 16$ .
Solve $\;4x - 3 = 9$ .
Solve $\;7a - 4 = 24$ .
Solve $\;\dfrac{x}{2} + 1 = 6$ .
Solve $\;\dfrac{m}{3} - 4 = 1$ .
Solve $\;\dfrac{x + 5}{2} = 4$ .
Solve $\;\dfrac{2x + 1}{3} = 5$ .
Solve $\;3(x + 2) = 18$ .

Reasoning

Verify and reason

Verify, without solving, whether $x = 6$ is a solution of $\;2x + 4 = 16$ .
Verify whether $y = 3$ is a solution of $\;5y - 1 = 15$ .
Zara solves $2x + 6 = 14$ by writing ” $2x = 14 + 6 = 20$ , so $x = 10$ ”. Explain her mistake and give the correct answer.
Find the missing number: solve $\;\square + 7 = 15$ and check by substitution.
Write an equation of your own whose solution is $x = 8$ , and verify it by substitution.

Problem solving

Worded problems

Four times a number, increased by $7$ , is $31$ . Find the number.
A pencil costs $2 and a ruler costs $r. Three pencils and a ruler cost $8 in total. Find $r$ .
Ava is $x$ years old. Her sister is $3$ years older. In $5$ years, the sum of their ages will be $33$ . How old is Ava now?
A taxi charges a $4 flag-fall and $2.50 per kilometre. A trip cost $29. How long was the trip?
The perimeter of an isosceles triangle with two equal sides of length $x$ cm and base $8$ cm is $30$ cm. Find $x$ .
Five consecutive natural numbers sum to $45$ . Find the smallest of them. (Hint: call it $n$ and write $n + (n+1) + \ldots$ )

Extension

Practice: Extension

Reasoning

Non-positive and two-sided equations

Solve $\;x + 11 = 5$ .
Solve $\;-2m = 14$ .
Solve $\;2x + 9 = 3$ .
Solve $\;-3x + 5 = 20$ .
Solve $\;10 - 2x = 4$ .
Solve $\;5x - 2 = 2x + 10$ .
Solve $\;4y + 7 = y + 19$ .
Solve $\;\dfrac{x}{2} + \dfrac{x}{3} = 10$ . (Hint: multiply both sides by $6$ .)
The sum of three consecutive integers is $-9$ . Find them.
A number increased by $40\%$ gives $84$ . What was the original number?

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 7 core - answers

Fluency

One-step equations

$x = 12$
$y = 19$
$x = 7$
$y = 21$
$x = 13$
$a = 10$
$m = 6$
$x = 24$

Fluency

Two-step equations

$x = 4$
$y = 4$
$x = 3$
$x = 3$
$a = 4$
$x = 10$
$m = 15$
$x = 3$ . Method: multiply by $2$ , $x + 5 = 8$ ; subtract $5$ .
$x = 7$ . Method: multiply by $3$ , $2x + 1 = 15$ ; then subtract $1$ and divide by $2$ .
$x = 4$ . Method: divide by $3$ , $x + 2 = 6$ .

Reasoning

Verify and reason

Yes. $2(6) + 4 = 12 + 4 = 16$ . ✓
No. $5(3) - 1 = 14$ , not $15$ .
Zara added $6$ when she should have subtracted. The inverse of " $+ 6$ " is " $- 6$ ". Correct working: $2x + 6 = 14 \Rightarrow 2x = 8 \Rightarrow x = 4$ .
$\square = 8$ . Check: $8 + 7 = 15$ . ✓
Many answers possible. Example: $3x - 4 = 20$ gives $x = 8$ . Check: $3(8) - 4 = 20$ . ✓

Problem solving

Worded problems

$6$ . Method: $4n + 7 = 31$ , so $4n = 24$ .
r = $2. Method: $3 \times 2 + r = 8$ , so $6 + r = 8$ .
Ava is $10$ . Method: Ava $= x$ ; sister $= x + 3$ . In $5$ years: $(x + 5) + (x + 8) = 33$ ; $2x + 13 = 33$ ; $x = 10$ .
$10$ km. Method: $4 + 2.50k = 29$ , so $2.50k = 25$ .
$x = 11$ cm. Method: $x + x + 8 = 30$ .
$7$ is the smallest. Method: $n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10 = 45$ ; $5n = 35$ ; $n = 7$ .

Extension - answers

Reasoning

Non-positive and two-sided equations

$x = -6$ . Method: subtract $11$ .
$m = -7$ . Method: divide by $-2$ .
$x = -3$ . Method: subtract $9$ ; divide by $2$ .
$x = -5$ . Method: subtract $5$ , $-3x = 15$ ; divide by $-3$ .
$x = 3$ . Method: subtract $10$ , $-2x = -6$ ; divide by $-2$ .
$x = 4$ . Method: subtract $2x$ from both sides, $3x - 2 = 10$ ; add $2$ , $3x = 12$ .
$y = 4$ . Method: subtract $y$ , $3y + 7 = 19$ ; subtract $7$ ; divide by $3$ .
$x = 12$ . Method: multiply by $6$ , $3x + 2x = 60$ ; $5x = 60$ .
$-4, -3, -2$ . Method: $3n + 3 = -9$ , so $n = -4$ .
$60$ . Method: $x \times 1.40 = 84$ , so $x = 60$ .

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