Algebraic Definitions and Substitution
This topic covers the fundamental vocabulary of algebra and the essential skill of substituting numerical values into expressions and formulae. Mastering these basics is crucial as they are the building blocks for solving more complex problems across all ESAT science and maths sections.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Strictly follow the order of operations (BIDMAS/BODMAS) when substituting values, especially with powers, roots, and negatives.
- An expression like '5x - 2' has no equals sign, while an equation '5x - 2 = 8' is only true for a specific value of x.
- An identity, often written with '≡', is a statement that is true for all possible values of its variables, for example, (x+1)2 ≡ x2 + 2x + 1.
- Be careful with signs when substituting negative numbers. Remember that (-a)2 = a2, but -a2 is negative.
- Factors can be numbers, single variables, or entire expressions in brackets, e.g., 'x' and '(x-4)' are both factors of x(x-4).
Formulae
V = (1/3) × π × r2 × h This is a typical scientific formula (volume of a cone) where you would substitute given values for the variables (radius 'r' and height 'h') to find a result. The process is the same for any formula you are given.
Definitions
- Expression
- A collection of terms, variables, and mathematical operators, but with no equals sign. Example: 3x2 - 7y.
- Equation
- A statement that two expressions are equal, which is only true for certain values of the variables. Example: 4x - 5 = 11.=
- Formula
- A specific type of equation that defines a relationship between different quantities. Example: F = ma.
- Identity
- An equation that is true for every possible value of its variables. Often denoted with '≡'. Example: sin2(θ) + cos2(θ) ≡ 1.
- Term
- A single component of an expression, separated by addition or subtraction. In '2ab - 5c', the terms are '2ab' and '-5c'.
- Factor
- A number or expression that divides another expression exactly, without a remainder. Example: (x+2) is a factor of x2 - 4.
Worked example
The 'drag force' F on a component is calculated using the formula F = k(v2 - u3) / (2d), where k is a constant, v and u are velocities, and d is a distance. Calculate F when k = 10, v = -5, u = 2, and d = 7.
- 1
Substitute the given numbers into the formula:
F = 10 × ((-5)2 - 23) / (2 × 7) - 2
First, evaluate the contents of the bracket, following order of operations.
Calculate the powers:
(-5)2 = 25 and 23 = 8 - 3
Complete the bracket calculation:
(25 - 8) = 17 - 4
The expression now is:
F = 10 × 17 / (2 × 7) - 5
Calculate the numerator:
10 × 17 = 170 - 6
Calculate the denominator:
2 × 7 = 14 - 7
Perform the final division:
F = 170 / 14Both are divisible by 2, simplifying to F = 85 / 7.
Answer: 85/7
Common mistakes
- ×Sign errors with negative numbers, particularly when raising to a power. For instance, incorrectly calculating (-5)2 as -25 instead of the correct +25.
- ×Violating the order of operations (BIDMAS). A common mistake would be to perform subtraction before powers inside a bracket.
- ×Errors in arithmetic multiplication or division, especially when dealing with larger numbers without a calculator.
- ×Confusing an expression with an equation and trying to 'solve' it without an equals sign being present.
No-calculator tips
- ✓When substituting into a fractional formula, fully evaluate the numerator and denominator as separate, smaller problems before attempting the final division.
- ✓Always simplify fractions by cancelling common factors before carrying out long division. It is easier to work with 85/7 than 170/14.
- ✓When multiplying a number by 10, simply add a zero to the end. When multiplying by 5, you can multiply by 10 and then halve the result (e.g., 17 × 5 = 170 / 2 = 85).