Forming and Solving Equations

Ann is 4 years older than Ben. Their ages add to 34. Form an equation and find Ben's age.

Solution:

• Name the unknown first, and choose the smaller quantity so the other is written in terms of it
• Let Ben's age be b
• Ann is 4 years older, so Ann's age is b + 4
• Their ages add to 34, which gives the equation b + (b + 4) = 34
• Collect: 2b + 4 = 34
• Subtract 4: 2b = 30, then divide by 2: b = 15
• Ben is 15 years old
• Answer the question asked: b = 15 alone is not an answer about ages, and checking gives 15 + 19 = 34

Three consecutive integers have a sum of 84. Find the integers.

Solution:

• Name the unknown first, and choose the one the others can be written from
• Let the smallest integer be n
• Consecutive means each is one more than the last, so the other two are n + 1 and n + 2
• Writing all three in terms of a single letter is what makes this solvable with one equation
• Their sum is 84, which gives n + (n + 1) + (n + 2) = 84
• Collect the three n terms and the two numbers: 3n + 3 = 84
• Subtract 3 from both sides: 3n = 81
• Divide by 3: n = 27
• The integers are 27, 28 and 29
• Answer the question rather than stopping at n = 27, and check: 27 + 28 + 29 = 84

A cyclist rides 20 km at x km/h and then a further 12 km at (x + 2) km/h. The whole ride takes 3 hours. Show that 3x² − 26x − 40 = 0, and find the speed for the first part of the ride.

Solution:

• Time is distance ÷ speed, so the two parts take 20/x hours and 12/(x + 2) hours
• The two times add up to the whole ride, so write that equation down before simplifying: 20/x + 12/(x + 2) = 3
• If one part had instead taken a stated time longer than the other, the times would be subtracted
• Multiply every term by x(x + 2): 20(x + 2) + 12x = 3x(x + 2)
• Expand: 32x + 40 = 3x² + 6x
• Collect every term on the right: 3x² − 26x − 40 = 0, the equation printed in the question
• Factorise: (3x + 4)(x − 10) = 0, so x = 10 or x = −4/3
• A speed cannot be negative, so x = 10 and the first part is ridden at 10 km/h
• Check: 20 ÷ 10 + 12 ÷ 12 = 2 + 1 = 3 hours

A rectangle has length (2x + 1) cm and width (x − 3) cm. Its perimeter is 38 cm. Find the length and the width.

Solution:

• The property of the shape supplies the equation, so name it before writing anything: a rectangle's perimeter is twice the length plus twice the width
• 2((2x + 1) + (x − 3)) = 38
• Simplify inside the bracket by collecting like terms: 2x + x = 3x, and 1 − 3 = −2
• 2(3x − 2) = 38
• Divide both sides by 2 rather than expanding, which keeps the numbers small: 3x − 2 = 19
• Add 2: 3x = 21, so x = 7
• x = 7 is not the answer: the question asks for two lengths
• Length = 2 × 7 + 1 = 15 cm, width = 7 − 3 = 4 cm
• Check against the perimeter given: 2(15 + 4) = 2 × 19 = 38 cm, and both sides are positive, which a length must be

The diagram shows a rectangle (x + 4) cm long and (x + 1) cm wide. A square of side 3 cm has been cut from one corner, and the shaded area that is left is 31 cm². Show that x² + 5x − 36 = 0.

Solution:

• The area fact supplies the equation: the rectangle's area minus the square's area equals the shaded area
• Write it with the brackets still in place, before expanding anything: (x + 4)(x + 1) − 3² = 31
• Expand the brackets: x² + x + 4x + 4 − 9 = 31
• Collect like terms: x² + 5x − 5 = 31
• Subtract 31 from both sides: x² + 5x − 36 = 0, the equation printed in the question
• The working runs from your own equation to the printed one with every line shown, which is what the last mark needs
• As a check, x = 4 satisfies it (16 + 20 − 36 = 0) and gives an 8 cm by 5 cm rectangle with 40 − 9 = 31 cm² shaded