(a) Write down an expression for the total mass of c cricket balls, each weighing 160 grams, and f footballs, each weighing 400 grams.
(b) Expand and simplify.
3(2x – 5y) – 4(x – 2y)
(c) Factorise completely.
5x2y – 20x
(d) Solve the simultaneous equations.
3x + 4y = 7
4x – 3y = 26
▶️ Answer/Explanation
(a) Ans: 160c + 400f
Total mass = (mass of cricket balls) + (mass of footballs). Each cricket ball weighs 160g, so total for c balls is 160c. Each football weighs 400g, so total for f footballs is 400f.
(b) Ans: 2x – 7y
First expand: 6x – 15y – 4x + 8y. Then combine like terms: (6x – 4x) + (–15y + 8y) = 2x – 7y.
(c) Ans: 5x(xy – 4)
Factor out the greatest common factor (GCF), which is 5x. This gives 5x(xy – 4).
(d) Ans: [x=] 5 [y=] –2
Multiply the first equation by 3 and the second by 4 to eliminate y: 9x + 12y = 21 and 16x – 12y = 104. Add them to get 25x = 125, so x = 5. Substitute x = 5 into the first equation to find y = –2.
(a) In 2001 Arnold was x years old.
Ken is 34 years younger than Arnold.
(i) Complete the table, in terms of x, for Arnold’s and Ken’s ages.
(ii) In 2013 Arnold is three times as old as Ken.
Write down an equation in x and solve it.
(b) Solve the simultaneous equations.
3x + 2y = 18
2x – y = 19
▶️ Answer/Explanation
(a) (i) Ans: Arnold (2013) = x + 12, Ken (2001) = x − 34, Ken (2013) = x − 22
From 2001 to 2013, 12 years pass. Arnold’s age becomes \( x + 12 \). Ken’s age in 2001 is \( x – 34 \), so in 2013 it is \( x – 34 + 12 = x – 22 \).
(a) (ii) Ans: x = 39
Set Arnold’s age in 2013 as 3 times Ken’s age: \( x + 12 = 3(x – 22) \). Solving gives \( x + 12 = 3x – 66 \), then \( 78 = 2x \), so \( x = 39 \).
(b) Ans: x = 8, y = -3
From the second equation, \( y = 2x – 19 \). Substitute into the first: \( 3x + 2(2x – 19) = 18 \). Simplify to \( 7x – 38 = 18 \), so \( x = 8 \). Then \( y = 2(8) – 19 = -3 \).