Linear equations - Answer key

Fluency

Fluency

$x = 4$
$y = 4$
$x = 3$
$x = 3$
$a = 4$
$x = 10$
$m = 15$
$x = 3$ . 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

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

$6$ . 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: $n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10 = 45$ ; $5n = 35$ ; $n = 7$ .

Reasoning

$x = -6$ . 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 $x = 60$ .

Year 7 core - answers