iGCSE Mathematics (0580) : 2.5 Derive and solve simple linear equations in one unknown. iGCSE Style Questions Paper 3

Question

(a) \( n = \frac{r}{15} + 20 \)

Find the value of \( r \) when \( n = 180 \) and \( c = 3 \).

(b) Factorise completely: \( 20p^2q – 5p \)

Write down an expression for the total cost, in cents, of \( x \) apples and \( y \) bananas.

(d) Solve the simultaneous equations:

\( 8x + 3y = 59 \)

\( 5x + 7y = 83 \)

▶️ Answer/Explanation

Solution

(a) Ans: 8

Substitute \( n = 180 \) and \( c = 3 \) into the equation: \( 180 = \frac{r}{15} + 20 \times 3 \).

Simplify to get \( 180 = \frac{r}{15} + 60 \), then solve for \( r = 8 \).

(b) Ans: \( 5p(4pq – 1) \)

Factor out the greatest common factor \( 5p \) from both terms.

(c) Ans: \( 35x + 14y \)

Multiply the cost of each apple (35 cents) by \( x \) and each banana (14 cents) by \( y \), then add them together.

(d) Ans: \( x = 4 \), \( y = 9 \)

Multiply the first equation by 7 and the second by 3 to eliminate \( y \).

Subtract the equations to find \( x = 4 \), then substitute back to find \( y = 9 \).

Question

(a) Expand and simplify.

(i) \( 4(x+3) + 2(x-1) \)

(ii) \((m-6)(m-4)\)

(b) Make \( t \) the subject of the formula \( p = 4t + 3 \).

The perimeter of this triangle is 49 cm.

Work out the value of \( x \).

▶️ Answer/Explanation

Solution

8(a)(i) Ans: \(6x + 10\)

First expand both terms: \(4x + 12 + 2x – 2\)

Then combine like terms: \(6x + 10\)

8(a)(ii) Ans: \(m^2 – 10m + 24\)

Use FOIL method: \(m \times m – 4m – 6m + 24\)

Simplify to get the final expression.

8(b) Ans: \(t = \frac{p-3}{4}\)

Subtract 3 from both sides: \(p – 3 = 4t\)

Then divide both sides by 4 to isolate \(t\).

8(c) Ans: \(x = 9\)

Add all sides: \(4x – 15 + 2x + 1 + x = 49\)

Combine terms: \(7x – 14 = 49\)

Solve for \(x\): \(7x = 63\) → \(x = 9\)

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