Year 7 Maths Algebra and Equations

Equation Problems

Q: What does subtracted from mean?

Start with the second number. 4y subtracted from 30 is 30 - 4y.

Q: How do you handle two unknowns?

Write both with one pronumeral, such as x and 3x.

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Equation Problems - Video - Solving Word Problem Using Equations Watch

Theory

To solve a word problem, turn the words into an equation, solve the equation, then answer the question in words. This Year 7 maths page sets out five steps for every problem and works through number puzzles, money, formulas, shapes, ages and sharing problems.

1. Define the pronumeral. Say what it stands for, with units: $n$ = the number of tickets, $l$ = the length in cm.

2. Write the equation. Turn the words into algebra one piece at a time. Words like is, gives, leaves and costs become $=$.

3. Solve it. Use inverse operations on both sides, one step per line.

4. Answer in words, with units. “$g = 12$” becomes “She played $12$ games.”

5. Check the answer in the words of the question, not just in your equation.

Formulas: write the formula, substitute the values you know, then solve for the one that is left. Two unknowns: write both with one pronumeral. If Sam is $4$ years older than Ava, use $x$ for Ava and $x + 4$ for Sam.

Pizzas at $\$14$ each plus $\$8$ delivery cost $\$120$: $14p + 8 = 120$. Undo: $120 - 8 = 112$, then $112 \div 14 = 8$ pizzas.

Words to equations:

Words	Equation
A number is doubled, then $5$ is added. The result is $19$.	$2n + 5 = 19$
$7$ is taken away from a number tripled, leaving $20$.	$3n - 7 = 20$
A number tripled is subtracted from $50$, giving $14$.	$50 - 3n = 14$
A $\$5$ booking fee plus $\$3$ per hour costs $\$26$.	$5 + 3h = 26$
A rectangle with width $6$ cm has a perimeter of $40$ cm.	$2l + 2 \times 6 = 40$
A rectangle $4$ cm wide and $(x + 3)$ cm long has an area of $28$ cm².	$4(x + 3) = 28$

          Same units throughout. Before writing an equation with \(1.5\) L and \(400\) mL, change \(1.5\) L to \(1500\) mL.
        

          Answer the question asked. If Ava is \(x = 9\) and the question asks for Sam’s age, the answer is \(x + 4 = 13\) years.
        

Five steps for every problem

Define the pronumeral, with units.
Write the equation from the words.
Solve it, one step per line.
Answer in words, with units.
Check in the words of the question.

Example 1 — Think of a number

A number is multiplied by $4$, then $9$ is subtracted. The result is $23$. What is the number?

Solution

Let $n$ be the number.

$4n - 9$	$=$	$23$
$4n$	$=$	$32$
$n$	$=$	$8$

4 n - 9 = 23, 4 n = 32, n = 8

The number is $8$. Check: $4 \times 8 - 9 = 23$.

Example 2 — Subtracted from

The product of $y$ and $4$ is subtracted from $30$ to give the answer $18$. What is the value of $y$?

Solution

“Subtracted from $30$” means start with $30$: $30 - 4y = 18$.

$-4y$	$=$	$18 - 30 = -12$
$y$	$=$	$\dfrac{-12}{-4} = 3$

30 - 4 y = 18, - 4 y = - 12, y = 3

$y = 3$. Check: $30 - 4 \times 3 = 18$.

Example 3 — A pizza order

Ms Nguyen orders pizzas for a class party. Each pizza costs $\$14$ and delivery is $\$8$. She pays $\$120$. How many pizzas did she order?

Solution

Let $p$ be the number of pizzas.

$14p + 8$	$=$	$120$
$14p$	$=$	$112$
$p$	$=$	$8$

14 p + 8 = 120, 14 p = 112, p = 8

She ordered $8$ pizzas. Check: $14 \times 8 + 8 = 120$.

Example 4 — Formulas

(i) The area of a triangle is $12$ cm² and the base is $3$ cm. Use $A = \dfrac{1}{2}bh$ to find the height.
(ii) When $t$ tables are joined in a row, the number of seats is $S = 4t + 2$. How many tables seat $42$ people?

Solution

$12$	$=$	$\dfrac{1}{2} \times 3 \times h$
$3h$	$=$	$24$
$h$	$=$	$8$
$4t + 2$	$=$	$42$
$t$	$=$	$40 \div 4 = 10$

12 = 1 / 2 \times 3 \times h, 3 h = 24, h = 84 t + 2 = 42, t = 10

(i) The height is $8$ cm.
(ii) $10$ tables

Example 5 — Two unknowns

Tom has $3$ times as many footy cards as Mia. Together they have $48$ cards. How many cards does each have?

Solution

Let $x$ be Mia’s cards; Tom has $3x$.

$x + 3x$	$=$	$48$
$4x$	$=$	$48$
$x$	$=$	$12$

x + 3 x = 48, 4 x = 48, x = 12

Mia has $12$ cards and Tom has $3 \times 12 = 36$. Check: $12 + 36 = 48$.

Example 6 — Shapes

(i) A rectangle with width $6$ cm has a perimeter of $40$ cm. Find its length.
(ii) A rectangle $4$ cm wide and $(x + 3)$ cm long has an area of $28$ cm². Find $x$.

Solution

$2l + 2 \times 6$	$=$	$40$
$2l$	$=$	$28$
$l$	$=$	$14$
$4(x + 3)$	$=$	$28$
$x + 3$	$=$	$7$
$x$	$=$	$4$

2 l + 12 = 40, 2 l = 28, l = 144 (x + 3) = 28, x + 3 = 7, x = 4

(i) $14$ cm
(ii) $x = 4$, so the rectangle is $7$ cm long.

Example 7 — Ages and units

(i) In $18$ years’ time, Sam will be $4$ times as old as he is now. How old is Sam?
(ii) A $1.5$ L jug holds $400$ mL of juice and the rest is water. How much water is in it?

Solution

$x + 18$	$=$	$4x$
$18$	$=$	$3x$
$x$	$=$	$6$
$400 + w$	$=$	$1500$
$w$	$=$	$1100$

x + 18 = 4 x, 18 = 3 x, x = 6400 + w = 1500, w = 1100

(i) Sam is $6$ (in $18$ years he will be $24 = 4 \times 6$).
(ii) Change $1.5$ L to $1500$ mL first: $1100$ mL of water.

Common pitfalls

Subtracted from in the wrong order. $3n$ subtracted from $50$ is $50 - 3n$, not $3n - 50$.

Mixing units. Change $1.5$ L to $1500$ mL before adding $400$ mL.

Answering with only the pronumeral. Write a sentence: “She ordered $8$ pizzas”, not just “$p = 8$”.

Finding only one of two unknowns. If the question asks for both people, find both and check both facts.

Frequently asked questions

How do you turn a word problem into an equation?

Define a pronumeral, then turn the words into algebra piece by piece. Words like is and costs become $=$.

What does subtracted from mean?

Start with the second number. $4y$ subtracted from $30$ is $30 - 4y$.

How do you use a formula to find an unknown?

Substitute the values you know, then solve the equation for the one that is left.

How do you handle two unknowns?

Write both with one pronumeral, such as $x$ and $3x$.

Why should you check in the words of the question?

Your equation might be wrong; checking in the story catches that.

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Words	Equation
A number is doubled, then \(5\) is added. The result is \(19\).	\(2n + 5 = 19\)
\(7\) is taken away from a number tripled, leaving \(20\).	\(3n - 7 = 20\)
A number tripled is subtracted from \(50\), giving \(14\).	\(50 - 3n = 14\)
A \(\$5\) booking fee plus \(\$3\) per hour costs \(\$26\).	\(5 + 3h = 26\)
A rectangle with width \(6\) cm has a perimeter of \(40\) cm.	\(2l + 2 \times 6 = 40\)
A rectangle \(4\) cm wide and \((x + 3)\) cm long has an area of \(28\) cm².	\(4(x + 3) = 28\)

\(4n - 9\)	\(=\)	\(23\)
\(4n\)	\(=\)	\(32\)
\(n\)	\(=\)	\(8\)

\(-4y\)	\(=\)	\(18 - 30 = -12\)
\(y\)	\(=\)	\(\dfrac{-12}{-4} = 3\)

\(14p + 8\)	\(=\)	\(120\)
\(14p\)	\(=\)	\(112\)
\(p\)	\(=\)	\(8\)

\(12\)	\(=\)	\(\dfrac{1}{2} \times 3 \times h\)
\(3h\)	\(=\)	\(24\)
\(h\)	\(=\)	\(8\)
\(4t + 2\)	\(=\)	\(42\)
\(t\)	\(=\)	\(40 \div 4 = 10\)

\(2l + 2 \times 6\)	\(=\)	\(40\)
\(2l\)	\(=\)	\(28\)
\(l\)	\(=\)	\(14\)
\(4(x + 3)\)	\(=\)	\(28\)
\(x + 3\)	\(=\)	\(7\)
\(x\)	\(=\)	\(4\)

\(x + 18\)	\(=\)	\(4x\)
\(18\)	\(=\)	\(3x\)
\(x\)	\(=\)	\(6\)
\(400 + w\)	\(=\)	\(1500\)
\(w\)	\(=\)	\(1100\)

Equation Problems

📖 Theory

Five steps for every problem

Common pitfalls

Frequently asked questions

Theory