Substitution Quizzes
Visual overview of Substitution.
Introduction
Substitution means replacing variables with given numerical values to calculate the result of an expression or formula. It is one of the most important skills in GCSE Maths and is used in algebra, geometry, physics, and finance. By mastering substitution, students can evaluate expressions, solve equations, and apply mathematical models to real-life problems accurately.
Example: If \(x=3\) and \(y=5\), then \(x+y=3+5=8\).
Core Concepts
What is Substitution?
Substitution replaces each variable with its numerical value, then simplifies using the correct order of operations (BIDMAS).
- \(2x+5,\;x=4 \Rightarrow 2(4)+5=13\)
- \(A=l\times w,\;l=7,\;w=3 \Rightarrow A=7\times3=21\)
Steps for Substitution
- Identify the variable(s) in the expression.
- Write the given values beside each variable.
- Substitute carefully, adding brackets if necessary.
- Apply BIDMAS to simplify step by step.
Substituting Multiple Variables
Replace each variable with its correct value and simplify in order.
Example
- Expression: \(3x+2y-z\)
- Values: \(x=2,\;y=4,\;z=5\)
- Substitute: \(3(2)+2(4)-5=6+8-5=9\)
Substituting Negative Numbers
Use brackets to preserve the correct sign.
Example
- Expression: \(x-y\)
- Values: \(x=5,\;y=-3\)
- Substitute: \(5-(-3)=5+3=8\)
Substituting Fractions
Convert fractions to a common denominator or decimals if easier, then simplify.
- \(x+y,\;x=\tfrac{1}{2},\,y=\tfrac{3}{4}\Rightarrow \tfrac{1}{2}+\tfrac{3}{4}=\tfrac{5}{4}\)
Substituting Decimals
Work as with whole numbers, applying standard arithmetic.
- \(0.5x+0.3y,\;x=4,\,y=3 \Rightarrow 0.5(4)+0.3(3)=2+0.9=2.9\)
Substitution with Indices
Follow BIDMAS: powers come before multiplication or addition.
- \(x^2+2x+1,\;x=3\Rightarrow 3^2+2(3)+1=9+6+1=16\)
Substitution in Real-Life Formulas
Substitution applies across geometry, physics, and finance formulas:
- Area of rectangle \(A=l\times w\)
- Speed \(s=\tfrac{d}{t}\)
- Force \(F=m\times a\)
- Simple Interest \(I=P\times r\times t\)
- Volume \(V=\pi r^2h\)
Worked Examples
Example 1 (Foundation): Single Variable
\(2x+5,\;x=3 \Rightarrow 2(3)+5=11\)
Example 2 (Foundation): Two Variables
\(x+y,\;x=4,\;y=5 \Rightarrow 4+5=9\)
Example 3 (Higher): Three Variables
\(3x-2y+z,\;x=2,\;y=4,\;z=5 \Rightarrow 3(2)-2(4)+5=3\)
Example 4 (Higher): Negative Numbers
\(x-y,\;x=5,\;y=-3 \Rightarrow 5-(-3)=8\)
Example 5 (Higher): Fractions
\(x+y,\;x=\tfrac{1}{3},\,y=\tfrac{2}{5}\Rightarrow\tfrac{11}{15}\)
Example 6 (Higher): Decimals
\(0.2x+0.5y,\;x=3,\,y=4 \Rightarrow 2.6\)
Example 7 (Higher): Indices
\(x^2+2x+1,\;x=4 \Rightarrow 4^2+8+1=25\)
Example 8 (Higher): Real-Life Formula (Speed)
\(s=\tfrac{d}{t},\;d=150,\;t=3 \Rightarrow 50\,\text{km/h}\)
Example 9 (Higher): Physics Formula
\(F=m\times a,\;m=10,\;a=-2 \Rightarrow F=-20\,\text{N}\)
Example 10 (Higher): Finance Formula
\(I=P\times r\times t,\;P=500,\,r=0.05,\,t=3 \Rightarrow I=75\)
Common Mistakes
- Forgetting to substitute all variables.
- Missing brackets around negatives.
- Breaking BIDMAS rules.
- Using percentages as whole numbers instead of decimals.
- Arithmetic slips with fractions or decimals.
Applications
- Geometry: finding area, perimeter, or volume.
- Physics: equations for motion, force, and energy.
- Finance: calculating interest, profit, and depreciation.
- Everyday maths: distance-time, budgeting, conversions.
Strategies & Tips
- Use brackets for clarity—especially with negatives.
- Follow BIDMAS carefully at every step.
- Convert fractions or percentages before substituting.
- Double-check all substitutions and arithmetic.
- Practise with multi-variable and real-world formulas.
Summary / Call-to-Action
Substitution transforms algebra from symbols into numbers. By replacing variables with values correctly—whether integers, fractions, or decimals—you can evaluate expressions, apply formulas, and solve problems accurately. It’s a skill used across all topics, from geometry to finance.
- Practise one-variable and multi-variable substitutions.
- Apply substitution in speed, area, and interest problems.
- Use brackets and BIDMAS to avoid sign errors.