Year 7 Maths Algebra and Equations

One Step Equations (add and subtract)

Q: How do you solve x + 15 = 23?

Subtract 15 from both sides: x = 8.

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Theory

To solve an equation, undo what has been done to the pronumeral. Use the inverse operation and do it to both sides. This Year 7 maths page shows the balancing method and the backtracking method for equations with $+$ and $-$, including negative answers and word problems.

An equation is a statement with a variable, numbers and an equals sign.

The balancing method. An equation is like a balance: both sides have the same value. Do the same operation to both sides and it stays balanced.

The backtracking method. Use a flow chart to show what was done to the variable, then undo each operation with its inverse (opposite) in reverse order.

Adding and subtracting are inverse operations: $x + 17 = 30$: subtract $17$ from both sides. $x - 4 = 9$: add $4$ to both sides.

Write one step per line, then check by substituting: LHS $=$ RHS.

$x + 5 = 12$ is a balance.

Take $5$ from both sides and it stays balanced: $x + 5 - 5 = 12 - 5$, so $x = 7$.

Backtracking $a - 6 = 9$: the variable had $6$ taken away. Undo it: $9 + 6 = 15$.

Undo with the inverse:

Equation	Do to both sides	Solution
$x + 15 = 23$	subtract $15$	$x = 8$
$x - 4 = 22$	add $4$	$x = 26$
$12 = b - 5$	add $5$	$b = 17$
$8 - x = 3$	add $x$, then subtract $3$	$x = 5$

          Collect the numbers first. A shape with sides \(x\), \(5\), \(8\) and \(6\) cm and a perimeter of \(27\) cm gives \(x + 19 = 27\), so \(x = 8\).
        

          Pronumeral on the right. Solve \(12 = b - 5\) the same way, then write \(b = 17\) with \(b\) on the left.
        

How to solve a one-step equation

Find what was done to the pronumeral ($+$ or $-$).
Do the inverse operation to both sides.
Simplify each side, one step per line.
Check by substituting into the original equation.

Example 1 — Two methods

Solve $a - 6 = 9$.

Solution

Balancing method: add $6$ to both sides.

$a - 6 + 6$	$=$	$9 + 6$
$a$	$=$	$15$

a - 6 + 6 = 9 + 6, a = 15

Backtracking method: undo “$- 6$” by adding $6$: $a = 9 + 6 = 15$. Check: $15 - 6 = 9$.

Example 2 — Undo adding or subtracting

Solve
(i) $x + 15 = 23$
(ii) $x - 4 = 22$

Solution

$x + 15 - 15$	$=$	$23 - 15$
$x$	$=$	$8$
$x - 4 + 4$	$=$	$22 + 4$
$x$	$=$	$26$

x + 15 - 15 = 23 - 15, x = 8 x - 4 + 4 = 22 + 4, x = 26

(i) $x = 8$. Check: $8 + 15 = 23$.
(ii) $x = 26$. Check: $26 - 4 = 22$.

Example 3 — A negative answer

Solve $x + 11 = 4$.

Solution

$x + 11 - 11$	$=$	$4 - 11$
$x$	$=$	$-7$

x + 11 - 11 = 4 - 11, x = - 7

$4 - 11$ goes $7$ below zero. Check: $-7 + 11 = 4$. So $x = -7$.

Example 4 — A cold night

Overnight in the mountains the temperature fell by $7^\circ$C to $-4^\circ$C. Let $t$ be the starting temperature in $^\circ$C. Write an equation and solve it.

Solution

$t - 7$	$=$	$-4$
$t - 7 + 7$	$=$	$-4 + 7$
$t$	$=$	$3$

t - 7 = - 4, t = - 4 + 7 = 3

It started at $3^\circ$C. That makes sense: it was warmer before it fell.

Example 5 — The pronumeral is subtracted

Solve $5 - x = -9$.

Solution

Add $x$ to both sides so $x$ is no longer subtracted: $5 = -9 + x$. Then add $9$ to both sides.

$5 + 9$	$=$	$x$
$x$	$=$	$14$

5 = - 9 + x, 5 + 9 = x, x = 14

Check: $5 - 14 = -9$. So $x = 14$.

Example 6 — Stories

(i) Zoe spends $\$12$ and has $\$35$ left. Let $m$ be the money she started with. Write and solve an equation.
(ii) A shape has sides $x$ cm, $5$ cm, $8$ cm and $6$ cm and a perimeter of $27$ cm. Find $x$.

Solution

$m - 12$	$=$	$35$
$m$	$=$	$35 + 12 = 47$
$x + 19$	$=$	$27$
$x$	$=$	$27 - 19 = 8$

m - 12 = 35, m = 47 x + 19 = 27, x = 8

(i) Zoe started with $\$47$.
(ii) $x = 8$ cm

Common pitfalls

Doing the same operation instead of the inverse. For $x - 6 = 10$, add $6$: $x = 16$, not $10 - 6 = 4$.

Losing the sign. For $x + 9 = 4$, $x = 4 - 9 = -5$, not $5$ or $13$.

Treating 8 - x like x - 8. In $8 - x = 3$, $x$ is subtracted. Add $x$ first: $8 = 3 + x$, so $x = 5$.

Changing one side only. Whatever you do to one side, do to the other side too.

Frequently asked questions

What is an inverse operation?

The operation that undoes another. Adding and subtracting are inverses.

How do you solve x + 15 = 23?

Subtract $15$ from both sides: $x = 8$.

What is the backtracking method?

Show what was done to the variable in a flow chart, then undo each step with its inverse in reverse order.

Can the answer to an equation be negative?

Yes. $x + 11 = 4$ gives $x = -7$.

How do you check your answer?

Substitute it into the original equation and check that both sides are equal.

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One Step Equations (multiply and divide)

Equation	Do to both sides	Solution
\(x + 15 = 23\)	subtract \(15\)	\(x = 8\)
\(x - 4 = 22\)	add \(4\)	\(x = 26\)
\(12 = b - 5\)	add \(5\)	\(b = 17\)
\(8 - x = 3\)	add \(x\), then subtract \(3\)	\(x = 5\)

\(a - 6 + 6\)	\(=\)	\(9 + 6\)
\(a\)	\(=\)	\(15\)

\(x + 15 - 15\)	\(=\)	\(23 - 15\)
\(x\)	\(=\)	\(8\)
\(x - 4 + 4\)	\(=\)	\(22 + 4\)
\(x\)	\(=\)	\(26\)

\(x + 11 - 11\)	\(=\)	\(4 - 11\)
\(x\)	\(=\)	\(-7\)

\(t - 7\)	\(=\)	\(-4\)
\(t - 7 + 7\)	\(=\)	\(-4 + 7\)
\(t\)	\(=\)	\(3\)

\(m - 12\)	\(=\)	\(35\)
\(m\)	\(=\)	\(35 + 12 = 47\)
\(x + 19\)	\(=\)	\(27\)
\(x\)	\(=\)	\(27 - 19 = 8\)

One Step Equations (add and subtract)

📖 Theory

How to solve a one-step equation

Common pitfalls

Frequently asked questions

Theory