Forming and solving linear equations

(a) $4x = 20$, so $x = 20 \div 4 = 5$

(b) $y = 24 - 7 = 17$

(c) $w = 4 \times 2 = 8$

Add 2 to both sides.

$p = 3 + 2 = 5$

Work backwards: $72 - 7 = 65$

$65 \div 5 = 13$

(a) $m - 3 = 4$, so $m = 4 + 3 = 7$.

(b) $3n + n = 4n$, so $4n = 24$ and $n = 24 \div 4 = 6$.

(a) Divide both sides by 3: $m = 36 \div 3 = 12$.

(b) $7 - x = 3$, so $x = 7 - 3 = 4$.

(a) $3t = 12$, so $t = 4$

(b) Add 2 to both sides: $x = 8$

(c) $6w + 2 = 20$

$6w = 18$

$w = 3$

(a) $3x = 51$, so $x = 51 \div 3 = 17$

(b) $\frac{y}{4} = 3$, so $y = 3 \times 4 = 12$

(c) $2f + 7 = 18 \Rightarrow 2f = 11 \Rightarrow f = 5.5$

(a) The highest common factor of 6 and 15 is 3: $3(2a + 5)$

(b) $4(3y + 1) = 28$

$3y + 1 = 7$

$3y = 6$, so $y = 2$

$2(4x - 5) = 18$

$4x - 5 = 9$

$4x = 14$

$x = 3.5$

$\frac{x}{7} + 9 = 4$

Subtract 9: $\frac{x}{7} = -5$

Multiply by 7: $x = -35$

$8x - 12 = 20$

$8x = 32$

$x = 4$

(a) $4c + 3c = 7c$ and $7d - d = 6d$, so $7c + 6d$.

(b) Divide by 5: $2m - 6 = 8$, so $2m = 14$ and $m = 7$.

(c) 3 boxes give $3x$ and 2 packets give $2y$: total $3x + 2y$.

(a) Multiply each term by 2: $2(a + d) = 2a + 2d$

(b) Both terms contain $y$: $6y^2 - 5y = y(6y - 5)$

(c) $4x - 7 = 37$

$4x = 44$

$x = 11$

(a) $x \times x - x \times 4 = x^2 - 4x$

(b) The highest common factor of 15 and 10 is 5: $5(3y - 2)$.

(c) Divide by 7: $f - 5 = 4$, so $f = 9$.

(a) $3x + 2x = 5x$ and $5y - 4y = y$, giving $5x + y$

(b) $5p = 22 - 7 = 15$, so $p = 15 \div 5 = 3$

Divide both sides by 4: $x - 6 = 11$

Add 6: $x = 17$

(a) $4c = 11 - 5 = 6$, so $c = 6 \div 4 = 1.5$

(b) $e + 7 = 20 \div 5 = 4$, so $e = 4 - 7 = -3$

(c) Multiply the powers: $(m^3)^2 = m^{3 \times 2} = m^6$

(a) $f + 2f + f = 4f$, so $4f = 20$ and $f = 5$.

(b) $18 - m = 6$ means $m = 18 - 6 = 12$.

(c) Add the powers: $d^2 \times d^3 = d^{2+3} = d^5$.

Add $x$ to both sides: $6x - 14 = 52$

Add 14 to both sides: $6x = 66$

$x = 11$

Perimeter: $(x + 1) + (x - 1) + 2x + (2x - 5) = 6x - 5$

$6x - 5 = 52$, so $6x = 57$ and $x = 9.5$

$DC = 2x = 2 \times 9.5 = 19$ cm

(a) $4 \times 0.5 - 2 = 0$, so AB would be 0 cm, which is impossible.

(b) A kite has two pairs of equal sides, so the perimeter is $2(AB + AD) = 2(AB + 3AB) = 8AB$.

$8AB = 64$, so $AB = 8$.

$4x - 2 = 8$, so $4x = 10$ and $x = 2.5$

(a) $3(x - 4) = 12 \Rightarrow x - 4 = 4 \Rightarrow x = 8$

(b) The highest common factor of $9b$ and $3b^2$ is $3b$.

$9b - 3b^2 = 3b(3 - b)$

(a) By symmetry $DE = AB = 2x + 1$ and $CD = BC = x + 2$

Perimeter $= 4x + 2(2x + 1) + 2(x + 2) = 4x + 4x + 2 + 2x + 4 = 10x + 6$

The perimeter is 18, so $10x + 6 = 18$

(b) $10x = 12$, so $x = 1.2$

The two unknown angles add to $90°$.

$7x + 5x + 18 = 90$

$12x = 72$, so $x = 6$

Angles: $7 \times 6 = 42°$ and $5 \times 6 + 18 = 48°$. The smallest is $42°$.

$x + 3x + 1 + 2x - 5 = 44$, so $6x - 4 = 44$, $6x = 48$, $x = 8$

Shelf B: $3 \times 8 + 1 = 25$ books, so one book is $7500 \div 25 = 300$ g

Shelf A: $8 \times 300 = 2400$ g

Triangle: $\frac{1}{2} \times 8 \times 6x = 24x$

Rectangle: $5(4x - 1) = 20x - 5$

$24x = 20x - 5 + 10$

$4x = 5$, so $x = 1.25$

(Check: triangle 30, rectangle 20.)

(a) $9 + x = 7(11 - x) = 77 - 7x$

$8x = 68$, so $x = 8.5$

(b) Cancel $(y + 3)^2$ from top and bottom: $\dfrac{4(y+3)^3}{(y+3)^2} = 4(y + 3)$

Area $= \frac{1}{2} \times 3x \times 3x = \frac{9x^2}{2}$.

$\frac{9x^2}{2} = 162$, so $9x^2 = 324$ and $x^2 = 36$.

$x = 6$.

Multiply both sides by 2: $5 - x = 4x - 14$.

Add $x$ and 14 to both sides: $19 = 5x$.

$x = \frac{19}{5} = 3.8$

Let Becky have $x$. Chris has $2x$ and Dan has $2x + 7$.

$x + 2x + 2x + 7 = 57$, so $5x = 50$ and $x = 10$

Becky 10, Chris 20, Dan 27

Equal shares would be $57 \div 3 = 19$ each, but Chris keeps his 20, so Dan is wrong.

Let square A have side $x$ cm, so square B has side $x + 4$.

$(x + 4)^2 = x^2 + 70$

$x^2 + 8x + 16 = x^2 + 70$, so $8x = 54$ and $x = 6.75$.

Side of B $= 10.75$ cm, so area $= 10.75^2 = 115.5625$.

Area of B $= 116$ cm² (3 s.f.)

Multiply every term by $6x$:

$2(x + 2) + 3(x - 2) = 18x$

$2x + 4 + 3x - 6 = 18x$

$5x - 2 = 18x$, so $-2 = 13x$ and $x = -\frac{2}{13}$.