Algebra: Equations Worksheet with Answers

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KS3 · Algebra

KS3.M-A2 Algebra: Equations

AQA AQA KS3 Maths - Algebra strand (non-statutory KS3 framework) · Calculator allowed · about 95 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Solve each equation.

(a)x + 7 = 15(1)

(b)x - 4 = 9(1)

(c)5x = 35(1)

(d)x/4 = 6(1)

2

Solve each equation, showing your working.

(a)3x + 4 = 19(2)

(b)5x - 3 = 22(2)

(c)2x + 7 = 1(2)

3

Solve each equation. Some solutions are negative.

(a)-2x + 5 = 11(2)

(b)10 - 3x = 25(2)

4

Solve each equation. Expand the bracket first.

(a)3(x + 2) = 21(2)

(b)4(2x - 1) = 28(2)

5

Solve each equation involving a fraction.

(a)x/3 + 5 = 9(2)

(b)x/5 - 2 = 3(2)

6

Tickets for a school concert cost L pounds each. Ffion buys 4 tickets and pays a booking fee of GBP3. The total she pays is GBP47.

(a)Write an equation in L for the total cost.(1)

(b)Solve your equation to find L.(2)

7

Solve each equation. The unknown appears on both sides.

(a)5x + 3 = 2x + 18(3)

(b)7x - 4 = 3x + 20(3)

8

A rectangle has length (x + 5) cm and width x cm. The perimeter of the rectangle is 46 cm.

Diagram NOT accurately drawn

(a)Write an equation in x for the perimeter of the rectangle.(1)

(b)Solve your equation to find x.(2)

(c)Hence find the length of the rectangle.(1)

9

Three consecutive integers have a sum of 72. Let the smallest integer be n.

(a)Write an equation in n for the sum of the three integers.(1)

(b)Solve your equation to find n.(2)

(c)State the three consecutive integers.(1)

10

The diagram shows three angles on a straight line: x degrees, 2x degrees and 3x degrees.

Diagram NOT accurately drawn

(a)Form an equation in x.(1)

(b)Solve your equation to find x.(2)

(c)Find the size of the largest angle.(1)

11

An isosceles triangle has two equal base angles of x degrees each. The apex angle is (x + 30) degrees.

Diagram NOT accurately drawn

(a)Form an equation in x, using the fact that angles in a triangle sum to 180 degrees.(1)

(b)Solve your equation to find x.(2)

(c)Find the size of the apex angle.(1)

12

Solve each equation. There are brackets on both sides.

(a)3(x + 4) = 2(x + 9)(3)

(b)5(2x - 3) = 3(3x + 1)(3)

13

Show that x = 7 is the solution of the equation 4(x - 2) = 2x + 6.

(2)

14

Solve the equation (x + 1)/2 = (x - 3)/3.

(3)

15

Solve the equation x² - 5 = 44, giving both solutions.

(3)

16

Solve the equation 6 - 2(x - 3) = 3x + 1.

(3)

17

Priya is x years old. Her brother Kofi is 3 years older than Priya. In 5 years' time, the sum of their ages will be 41.

(a)Write an expression, in terms of x, for Kofi's age now.(1)

(b)Form an equation in x using the information about the sum of their ages in 5 years' time.(1)

(c)Solve your equation to find x.(2)

(d)Find Kofi's current age.(1)

18

A number, n, is multiplied by 3, then 7 is added. The result is the same as when 10 is subtracted from double the number.

(a)Form an equation in n.(2)

(b)Solve your equation to find n.(2)

19

Solve the equation (3x - 1)/4 + (x + 5)/2 = 6, giving your answer as an integer.

(3)

20

The length of a rectangle is 3 cm more than twice its width, w. The perimeter of the rectangle is 54 cm.

Diagram NOT accurately drawn

(a)Write an expression for the length in terms of w.(1)

(b)Form an equation in w for the perimeter.(1)

(c)Solve your equation to find w.(2)

(d)Calculate the area of the rectangle.(1)

Mark scheme · KS3.M-A2 Algebra: Equations

Question 1

(a) B1 x = 8 (cao)
(a) Answer: x = 8
(b) B1 x = 13 (cao)
(b) Answer: x = 13
(c) B1 x = 7 (cao)
(c) Answer: x = 7
(d) B1 x = 24 (cao)
(d) Answer: x = 24

Question 2

(a) M1 correct rearrangement, 3x = 15 oe
(a) A1 x = 5 (cao)
(a) Answer: x = 5
(b) M1 correct rearrangement, 5x = 25 oe
(b) A1 x = 5 (cao)
(b) Answer: x = 5
(c) M1 correct rearrangement, 2x = -6 oe
(c) A1 x = -3 (cao)
(c) Answer: x = -3

Question 3

(a) M1 -2x = 6 oe
(a) A1 x = -3 (cao)
(a) Answer: x = -3
(b) M1 -3x = 15 oe
(b) A1 x = -5 (cao)
(b) Answer: x = -5

Question 4

(a) M1 correct expansion, 3x + 6 = 21 oe
(a) A1 x = 5 (cao)
(a) Answer: x = 5
(b) M1 correct expansion, 8x - 4 = 28 oe
(b) A1 x = 4 (cao)
(b) Answer: x = 4

Question 5

(a) M1 x/3 = 4 oe
(a) A1 x = 12 (cao)
(a) Answer: x = 12
(b) M1 x/5 = 5 oe
(b) A1 x = 25 (cao)
(b) Answer: x = 25

Question 6

(a) B1 4L + 3 = 47 oe
(a) Answer: 4L + 3 = 47
(b) M1 4L = 44 oe
(b) A1 L = 11 (cao)
(b) Answer: L = 11

Question 7

(a) M1 collects x terms on one side, 3x + 3 = 18 oe
(a) dM1 collects constants, 3x = 15 oe
(a) A1 x = 5 (cao)
(a) Answer: x = 5
(b) M1 collects x terms on one side, 4x - 4 = 20 oe
(b) dM1 collects constants, 4x = 24 oe
(b) A1 x = 6 (cao)
(b) Answer: x = 6

Question 8

(a) B1 2(x + x + 5) = 46 oe, e.g. 4x + 10 = 46
(a) Answer: 2(x + x + 5) = 46
(b) M1 4x = 36 oe
(b) A1 x = 9 (cao)
(b) Answer: x = 9
(c) B1 14 cm (ft their x + 5)
(c) Answer: 14 cm

Question 9

(a) B1 n + (n + 1) + (n + 2) = 72 oe
(a) Answer: n + (n + 1) + (n + 2) = 72
(b) M1 3n + 3 = 72 => 3n = 69 oe
(b) A1 n = 23 (cao)
(b) Answer: n = 23
(c) B1 23, 24, 25 (ft their n)
(c) Answer: 23, 24, 25

Question 10

(a) B1 x + 2x + 3x = 180 oe
(a) Answer: x + 2x + 3x = 180
(b) M1 6x = 180 oe
(b) A1 x = 30 (cao)
(b) Answer: x = 30
(c) B1 90 degrees (ft their x, largest = 3x)
(c) Answer: 90 degrees

Question 11

(a) B1 x + x + (x + 30) = 180 oe
(a) Answer: x + x + (x + 30) = 180
(b) M1 3x + 30 = 180 => 3x = 150 oe
(b) A1 x = 50 (cao)
(b) Answer: x = 50
(c) B1 80 degrees (ft their x + 30)
(c) Answer: 80 degrees

Question 12

(a) M1 expands both brackets correctly, 3x + 12 = 2x + 18
(a) dM1 collects terms correctly, x = 6 route, e.g. x + 12 = 18
(a) A1 x = 6 (cao)
(a) Answer: x = 6
(b) M1 expands both brackets correctly, 10x - 15 = 9x + 3
(b) dM1 collects terms correctly, x = 18 route, e.g. x - 15 = 3
(b) A1 x = 18 (cao)
(b) Answer: x = 18

Question 13

M1 substitutes x = 7 correctly into both sides, LHS = 4(5) = 20, RHS = 2(7) + 6 = 20
A1 states LHS = RHS = 20, so x = 7 is confirmed as the solution (cso)
Answer: LHS = RHS = 20, so x = 7 is the solution (shown)

Question 14

M1 cross-multiplies correctly, 3(x + 1) = 2(x - 3)
M1 expands correctly, 3x + 3 = 2x - 6
A1 x = -9 (cao)
Answer: x = -9

Question 15

M1 x² = 49 oe
A1 x = 7
A1 x = -7 (both roots required, cao)
Answer: x = 7 or x = -7

Question 16

M1 expands bracket correctly, 6 - 2x + 6 = 3x + 1, i.e. 12 - 2x = 3x + 1
dM1 rearranges correctly, 11 = 5x oe
A1 x = 2.2 oe (awrt 2.2)
Answer: x = 2.2

Question 17

(a) B1 x + 3
(a) Answer: x + 3
(b) B1 (x + 5) + (x + 3 + 5) = 41 oe, e.g. (x+5)+(x+8)=41
(b) Answer: (x + 5) + (x + 8) = 41
(c) M1 2x + 13 = 41 => 2x = 28 oe
(c) A1 x = 14 (cao)
(c) Answer: x = 14
(d) B1 17 (ft their x + 3)
(d) Answer: 17 years old

Question 18

(a) B1 3n + 7 for the left-hand side
(a) B1 3n + 7 = 2n - 10 oe, correct full equation
(a) Answer: 3n + 7 = 2n - 10
(b) M1 collects terms correctly, n = -17 route, e.g. n = -10 - 7
(b) A1 n = -17 (cao)
(b) Answer: n = -17

Question 19

M1 clears fractions correctly by multiplying every term by 4, (3x - 1) + 2(x + 5) = 24
dM1 expands and collects like terms correctly, 5x + 9 = 24
A1 x = 3 (cao)
Answer: x = 3

Question 20

(a) B1 2w + 3
(a) Answer: 2w + 3
(b) B1 2(w + 2w + 3) = 54 oe, e.g. 6w + 6 = 54
(b) Answer: 2(w + 2w + 3) = 54
(c) M1 6w = 48 oe
(c) A1 w = 8 (cao)
(c) Answer: w = 8
(d) B1 152 cm² (ft their w, length = 2w+3 = 19, area = 8 x 19)
(d) Answer: 152 cm²

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