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# GCSE Maths | Algebra
- URL: https://www.a-level-maths-tutor.com/gcse-maths-algebra/
- Published: 2026-08-05T15:30:49.000Z
- Updated: 2026-08-05T15:30:49.000Z
- Author: Jaisul Naik
- Tags: GCSE Maths

# GCSE Maths | Algebra

Collecting like terms

$$3a + 2b + 4a + b$$ $$= (3a+4a) + (2b+b)$$ $$= 7a + 3b$$ 

### Question 1

(a) $3a+2b+4a+b$

(b) $7y+5y+2h+2h$

(c) $g+8a+2a+g$

(d) $7m+7p+8m+p+2p$

(e) $9e+2+e+2$

(f) $4+3a+2a+8$

(g) $2y+4+3y-1$

(h) $8+3w-w-3$

(i) $5-4s-2+10s$

(j) $3x+6y+5x-2y$

(k) $6m-2s+11s+m$

(l) $2a+3b-2+a+3b+4$

Show Answers 

(a) $7a+3b$

(b) $12y+4h$

(c) $10a+2g$

(d) $15m+10p$

(e) $10e+4$

(f) $5a+12$

(g) $5y+3$

(h) $2w+5$

(i) $6s+3$

(j) $8x+4y$

(k) $7m+9s$

(l) $3a+6b+2$

### Question 2

(a) $3y^2+4ab+7y^2+ab$

(b) $9x^2-2x-11x^2+5x$

(c) $7ac-3ab+9ab-7ac$

Show Answers 

(a) $10y^2+5ab$

(b) $-2x^2+3x$

(c) $6ab$

Expanding brackets

$$5(y+3)$$ $$= 5 \\times y + 5 \\times 3$$ $$= 5y+15$$ 

### Question 3

(a) $5(y+3)$

(b) $4(a+2)$

(c) $8(w+10)$

(d) $3(x-7)$

(e) $9(s-1)$

(f) $2(8-t)$

(g) $7(4+h)$

(h) $10(a+2b+3c)$

Show Answers 

(a) $5y+15$

(b) $4a+8$

(c) $8w+80$

(d) $3x-21$

(e) $9s-9$

(f) $16-2t$

(g) $7h+28$

(h) $10a+20b+30c$

A minus sign outside the bracket flips the sign of every term inside it. $$-2(w+5)$$ $$= -2 \\times w + (-2) \\times 5$$ $$= -2w-10$$ 

### Question 4

(a) $-2(w+5)$

(b) $-3(c+7)$

(c) $-8(c+7)$

(d) $-10(y-2)$

(e) $-7(g-3)$

(f) $-4(2w+3)$

(g) $-9(3w-5)$

(h) $-9(5x-1)$

Show Answers 

(a) $-2w-10$

(b) $-3c-21$

(c) $-8c-56$

(d) $20-10y$

(e) $21-7g$

(f) $-8w-12$

(g) $45-27w$

(h) $9-45x$

### Question 5

(a) $a(a+2)$

(b) $y(y-5)$

(c) $w(a+w)$

(d) $c(9-c)$

(e) $p(2p+5)$

(f) $2w(3w-1)$

(g) $9y(2y+3)$

(h) $4c(2a+5c)$

Show Answers 

(a) $a^2+2a$

(b) $y^2-5y$

(c) $aw+w^2$

(d) $9c-c^2$

(e) $2p^2+5p$

(f) $6w^2-2w$

(g) $18y^2+27y$

(h) $8ac+20c^2$

Expand each bracket separately first, then collect like terms. $$5(y+3)+2(y+7)$$ $$= (5y+15)+(2y+14)$$ $$= 7y+29$$ 

### Question 6

(a) $5(y+3)+2(y+7)$

(b) $6(2w+5)+9(w+2)$

(c) $3(y-2)+4(2y+5)$

(d) $7(2g+3)-5(g+2)$

(e) $6(x-2)-4(x-8)$

(f) $2(3y-8)-5(2y-1)$

(g) $8(5+2m)+3(5-3m)$

(h) $4(w+7)-2(2w+1)$

(i) $9(1+2y)+3(3-y)$

Show Answers 

(a) $7y+29$

(b) $21w+48$

(c) $11y+14$

(d) $9g+11$

(e) $2x+20$

(f) $-4y-11$

(g) $7m+55$

(h) $26$

(i) $15y+18$

Multiply every term in the first bracket by every term in the second. $$(2c+1)(2c+3)$$ $$= 2c(2c+3) + 1(2c+3)$$ $$= 4c^2+6c+2c+3$$ $$= 4c^2+8c+3$$ 

### Question 7

(a) $(2c+1)(2c+3)$

(b) $(5x+1)(2x+5)$

(c) $(3w+2)(w+1)$

(d) $(3p+2)(2p-1)$

(e) $(5g-4)(g+1)$

(f) $(2a-3)(4a+7)$

Show Answers 

(a) $4c^2+8c+3$

(b) $10x^2+27x+5$

(c) $3w^2+5w+2$

(d) $6p^2+p-2$

(e) $5g^2+g-4$

(f) $8a^2+2a-21$

Factorising expressions

$$x^2+7x$$ The highest common factor of $x^2$ and $7x$ is $x$. $$= x(x+7)$$ 

### Question 8

(a) $x^2+7x$

(b) $x^2-3x$

(c) $y^2+y$

(d) $w^2+9w$

(e) $x^2-7x$

(f) $4w^2+10w$

(g) $6x^2-8x$

(h) $9y^2-6y$

(i) $10c+c^2$

(j) $5g-g^2$

(k) $14x^2+35x$

(l) $40x^2-50x$

(m) $12x^2+18x$

(n) $24x^2-18x$

(o) $45y^2+60y$

(p) $7w^2+2w$

Show Answers 

(a) $x(x+7)$

(b) $x(x-3)$

(c) $y(y+1)$

(d) $w(w+9)$

(e) $x(x-7)$

(f) $2w(2w+5)$

(g) $2x(3x-4)$

(h) $3y(3y-2)$

(i) $c(c+10)$

(j) $g(5-g)$

(k) $7x(2x+5)$

(l) $10x(4x-5)$

(m) $6x(2x+3)$

(n) $6x(4x-3)$

(o) $15y(3y+4)$

(p) $w(7w+2)$

### Question 9

(a) $x^2+xy$

(b) $a^2-ab$

(c) $xy+xz$

(d) $ab+ac-ad$

(e) $6c^2-4cd$

(f) $10x^2+15xy$

(g) $12ab+18bc$

(h) $8xy+4y^2$

(i) $8cdf+10cde$

(j) $7w^2+6w+wy$

(k) $8ab^2-10ab$

(l) $4xy^2+6xy+2x^2y$

(m) $6mn-7m^2n$

(n) $11g^2h+22h^2$

Show Answers 

(a) $x(x+y)$

(b) $a(a-b)$

(c) $x(y+z)$

(d) $a(b+c-d)$

(e) $2c(3c-2d)$

(f) $5x(2x+3y)$

(g) $6b(2a+3c)$

(h) $4y(2x+y)$

(i) $2cd(4f+5e)$

(j) $w(7w+6+y)$

(k) $2ab(4b-5)$

(l) $2xy(2y+3+x)$

(m) $mn(6-7m)$

(n) $11h(g^2+2h)$

### Question 10

(a) $x^3+2x^2$

(b) $5x^3-x^2$

(c) $8c^3+12c$

(d) $10w^2-15w^3$

(e) $32y^3+24y^2$

(f) $12x^4+15x$

(g) $4a^5-12a^2$

(h) $8w^9+w^7$

Show Answers 

(a) $x^2(x+2)$

(b) $x^2(5x-1)$

(c) $4c(2c^2+3)$

(d) $5w^2(2-3w)$

(e) $8y^2(4y+3)$

(f) $3x(4x^3+5)$

(g) $4a^2(a^3-3)$

(h) $w^7(8w^2+1)$

Rearranging equations

$$y+7=x$$ Subtract $7$ from both sides: $$y = x-7$$ 

### Question 11

a) $y+7=x$

b) $y+m=a$

c) $y+p=q$

d) $y+k=t$

e) $y+\\frac{1}{2}=a$

f) $y+b=x-c$

g) $y+n=2m$

h) $y+d=a+b$

i) $y+r=s-t$

j) $y+f=g-h$

Show Answers 

a) $y=x-7$

b) $y=a-m$

c) $y=q-p$

d) $y=t-k$

e) $y=a-\\frac{1}{2}$

f) $y=x-b-c$

g) $y=2m-n$

h) $y=a+b-d$

i) $y=s-t-r$

j) $y=g-h-f$

### Question 12

a) $y-5=x$

b) $y-n=m$

c) $y-c=p$

d) $y-t=a+b$

e) $y-d=\\frac{x}{2}$

f) $y-k=3m$

g) $y-r=s$

h) $y-(a+b)=c$

i) $y-p=q-r$

j) $y-f=g+h$

Show Answers 

a) $y=x+5$

b) $y=m+n$

c) $y=p+c$

d) $y=a+b+t$

e) $y=\\frac{x}{2}+d$

f) $y=3m+k$

g) $y=s+r$

h) $y=a+b+c$

i) $y=q-r+p$

j) $y=g+h+f$

$$4y=x$$ Divide both sides by $4$: $$y = \\frac{x}{4}$$ 

### Question 13

a) $4y=x$

b) $ay=b$

c) $ty=m$

d) $-3y=p$

e) $ky=a+b$

f) $ny=x-c$

g) $7y=3a$

h) $\\pi y=r$

i) $(a+b)y=c$

j) $my=n(p+q)$

Show Answers 

a) $y=\\frac{x}{4}$

b) $y=\\frac{b}{a}$

c) $y=\\frac{m}{t}$

d) $y=-\\frac{p}{3}$

e) $y=\\frac{a+b}{k}$

f) $y=\\frac{x-c}{n}$

g) $y=\\frac{3a}{7}$

h) $y=\\frac{r}{\\pi}$

i) $y=\\frac{c}{a+b}$

j) $y=\\frac{n(p+q)}{m}$

$$\\frac{y}{5}=x$$ Multiply both sides by $5$: $$y = 5x$$ 

### Question 14

a) $\\frac{y}{5}=x$

b) $\\frac{y}{k}=a$

c) $\\frac{y}{m}=p$

d) $\\frac{y}{a+b}=t$

e) $\\frac{y}{r}=s-t$

f) $\\frac{y}{2}=3x$

g) $\\frac{y}{d}=\\frac{a}{b}$

h) $\\frac{y}{\\pi}=r^2$

i) $\\frac{y}{f}=g+h$

j) $\\frac{y}{n}=m(k-t)$

Show Answers 

a) $y=5x$

b) $y=ak$

c) $y=mp$

d) $y=t(a+b)$

e) $y=r(s-t)$

f) $y=6x$

g) $y=\\frac{ad}{b}$

h) $y=\\pi r^2$

i) $y=f(g+h)$

j) $y=mn(k-t)$

$$2y+8=x$$ Subtract $8$, then divide by $2$: $$2y = x-8 \\implies y = \\frac{x-8}{2} = \\frac{x}{2}-4$$ 

### Question 15

a) $2y+8=x$

b) $ay-b=c$

c) $3y+k=m$

d) $-4y+p=q$

e) $\\frac{y}{2}+6=a$

f) $ny+t=x$

g) $ry-s=t$

h) $2y-b=a+b$

i) $\\frac{y}{3}+d=e$

j) $-y+a=b-c$

Show Answers 

a) $y=\\frac{x}{2}-4$

b) $y=\\frac{b+c}{a}$

c) $y=\\frac{m-k}{3}$

d) $y=\\frac{p-q}{4}$

e) $y=2a-12$

f) $y=\\frac{x-t}{n}$

g) $y=\\frac{s+t}{r}$

h) $y=\\frac{a}{2}+b$

i) $y=3(e-d)$

j) $y=a-b+c$

$$4(y+2)=x$$ Divide by $4$ first, then subtract $2$: $$y+2 = \\frac{x}{4} \\implies y = \\frac{x}{4}-2$$ 

### Question 16

a) $4(y+2)=x$

b) $a(y-b)=c$

c) $m(y+n)=p$

d) $-2(y+k)=t$

e) $\\frac{y+a}{b}=c$

f) $r(y-s)=t$

g) $y-d=\\frac{x}{k}$

h) $5(y-m)=n(p+q)$

i) $\\frac{y-t}{r}=s$

j) $a(y+b)=c(y-d)$

Show Answers 

a) $y=\\frac{x}{4}-2$

b) $y=\\frac{c}{a}+b$

c) $y=\\frac{p}{m}-n$

d) $y=-\\frac{t}{2}-k$

e) $y=bc-a$

f) $y=\\frac{t}{r}+s$

g) $y=\\frac{x}{k}+d$

h) $y=m+\\frac{n(p+q)}{5}$

i) $y=rs+t$

j) $y=\\frac{ab+cd}{c-a}$

$$-y=x$$ Divide both sides by $-1$ (i.e. flip every sign): $$y = -x$$ 

### Question 17

a) $-y=x$

b) $-y=a$

c) $a-y=b$

d) $-y+c=d$

e) $m-y=n$

f) $x+y=-y$

g) $p-y=q+r$

h) $-y=\\frac{a}{b}$

i) $k(-y+m)=n$

j) $a-y=b+y$

Show Answers 

a) $y=-x$

b) $y=-a$

c) $y=a-b$

d) $y=c-d$

e) $y=m-n$

f) $y=-\\frac{x}{2}$

g) $y=p-q-r$

h) $y=-\\frac{a}{b}$

i) $y=m-\\frac{n}{k}$

j) $y=\\frac{a-b}{2}$

When $y$ is in the denominator, multiply both sides by $y$ first, then divide by whatever's on the other side. $$\\frac{5}{y}=x$$ $$5 = xy \\implies y = \\frac{5}{x}$$ 

### Question 18

a) $\\frac{5}{y}=x$

b) $\\frac{a}{y}=b$

c) $\\frac{m}{y}=n$

d) $\\frac{p+q}{y}=r$

e) $\\frac{k}{y}=t$

f) $\\frac{a}{y}=b+c$

g) $\\frac{m}{y}=\\frac{n}{p}$

h) $\\frac{r}{y}=\\frac{s}{t}$

i) $\\frac{1}{y}=a$

j) $\\frac{a+b}{y}=c(d-e)$

Show Answers 

a) $y=\\frac{5}{x}$

b) $y=\\frac{a}{b}$

c) $y=\\frac{m}{n}$

d) $y=\\frac{p+q}{r}$

e) $y=\\frac{k}{t}$

f) $y=\\frac{a}{b+c}$

g) $y=\\frac{mp}{n}$

h) $y=\\frac{rt}{s}$

i) $y=\\frac{1}{a}$

j) $y=\\frac{a+b}{c(d-e)}$

If $y$ appears on both sides, get all the $y$ terms on one side and everything else on the other, then factorise out $y$ before dividing. $$3y+y=x$$ $$4y=x \\implies y=\\frac{x}{4}$$ 

### Question 19

a) $3y+y=x$

b) $ay-by=c$

c) $ay=by+c$

d) $2y+a-y=x$

e) $3y+k=y+m$

f) $py+qy=r$

g) $a(y+b)=cy+d$

h) $y-m=2y+n$

i) $\\frac{y}{a}=y-b$

j) $f(y+g)=h(y-j)$

Show Answers 

a) $y=\\frac{x}{4}$

b) $y=\\frac{c}{a-b}$

c) $y=\\frac{c}{a-b}$

d) $y=x-a$

e) $y=\\frac{m-k}{2}$

f) $y=\\frac{r}{p+q}$

g) $y=\\frac{d-ab}{a-c}$

h) $y=-m-n$

i) $y=\\frac{ab}{a-1}$

j) $y=\\frac{fg+hj}{h-f}$

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