MathematicsExam code: 0580
Algebra Toolkit
Algebra and Sequences
Replacing letters with numbers
- Substitution means putting given numbers in place of the letters, then evaluating
- Write the expression or formula down first, then replace each letter, then calculate
- Follow the order of operations exactly as with any numerical calculation
Negative values need brackets
- A negative value substituted for a letter is written in brackets, which keeps the sign attached through powers and multiplications
- Where x = −4, x² is (−4)² = 16, not −16
- Without brackets a calculator applies the power before the sign, giving the wrong result
Worked example
Substituting a negative value
Given a = 5 and b = −3, find the value of (i) 3a − b² (ii) 2b(a + b).
Solution:
- Substitute a negative value inside brackets every time, so the sign cannot be separated from the number
- (i) The expression becomes 3 × 5 − (−3)²
- BIDMAS puts the index before the subtraction, so square first
- (−3)² = (−3) × (−3) = 9, positive because two negatives multiply to a positive
- 15 − 9 = 6
- Without the bracket this reads 3 × 5 − −3², and since the power is applied before the sign, −3² is −9 and the answer comes out as 15 + 9 = 24, which is wrong
- (ii) The expression becomes 2 × (−3) × (5 + (−3))
- Brackets first: 5 + (−3) is 5 − 3 = 2
- Then left to right: 2 × (−3) = −6, and −6 × 2 = −12
- The answer is negative because exactly one of the factors is negative
Exam tip
Write the substituted line before working it out
Substitution is set in 7 of the 21 papers. Write the expression with the numbers in place before you work anything out, as in 3 × 5 − (−3)² above: in every two-mark version that line earns the method mark, so a slip in the arithmetic still keeps one mark. Three schemes also give that mark for a single piece worked out correctly, such as (−3)² = 9, so write each piece on its own line.
Worked example
Finding a letter that is squared in a formula
In the formula E = 2v² − u, find v when E = 58 and u = −8.
Solution:
- Substitute the values you know, with the negative in brackets: 58 = 2v² − (−8)
- Subtracting −8 is the same as adding 8, so 58 = 2v² + 8
- Take 8 from both sides: 2v² = 50, then divide by 2: v² = 25
- The formula gives v², so finish with a square root: v = 5 or −5, which can be written ±5
- Both values fit, because (−5)² = 25 as well; when a question asks for the positive value, give only 5