MathematicsExam code: 0580
Forming and Solving Equations
Algebra and Sequences
Building the statement
- An equation needs two expressions for the same quantity, set equal
- Read the sentence for the fact that links them: a total, a perimeter, an angle sum, or two costs being equal
- Check the expression before solving by substituting a trial value
Worked example
Consecutive integers
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
Worked example
Showing that journey times give a quadratic
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