Substitution (HSC SSCE Mathematics Standard): Revision Notes
Substitution
What is substitution?
Substitution is the process of putting numbers in place of letters (called pronumerals or variables) in an algebraic expression. Once you've replaced the letters with numbers, you work out the value of the expression and give your answer to the required level of accuracy.
This is a fundamental skill in algebra that you'll use throughout mathematics. You might need to substitute values into formulas for area, perimeter, interest calculations, or scientific equations.
Understanding substitution is essential for working with formulas in physics, chemistry, economics, and many other subjects. It's one of the most practical applications of algebra you'll encounter.
The substitution process
When substituting values into an algebraic expression, follow these steps carefully:
- Write the algebraic expression - Start by clearly writing out the expression you're working with.
- Replace the variables - Substitute each letter with its given number value. Put the numbers in brackets to avoid mistakes with negative numbers and operations.
- Evaluate - Calculate the result using your calculator. Remember to follow the correct order of operations (BIDMAS/BODMAS).
- Write your answer - Express your final answer to the specified level of accuracy and include the correct units if necessary.
Exam tip: Always put substituted values in brackets, especially negative numbers. This helps prevent sign errors and is one of the most common ways to lose marks in exams.
Working with linear expressions
A linear expression contains variables with a power of 1 (or no power shown). These expressions involve basic operations like addition, subtraction, and multiplication.
Worked Example: Evaluating a Linear Expression
Question: Evaluate , given , , and .
Solution:
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Write the expression:
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Substitute the values:
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Calculate step by step:
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Final answer:
Notice how we put brackets around each substituted value. This is especially important for to ensure we're adding a negative number correctly.
Working with non-linear expressions
Non-linear expressions contain variables with powers (like or ) or other operations like square roots. These require extra care when substituting and calculating.
Worked Example: Expression with a Cube
Question: Evaluate , given .
Solution:
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Write the expression:
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Substitute the value:
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Calculate the power first:
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Multiply:
Remember that means , so . The power operation happens before multiplication according to the order of operations.
Worked Example: Expression with a Square Root
Question: Evaluate , given .
Solution:
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Write the expression:
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Substitute the value:
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Work out what's under the square root:
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Take the square root:
Always complete all operations inside the square root symbol before taking the square root. The square root acts like a bracket, so you must evaluate everything inside it first.
Expressions with multiple operations
When an expression contains several different operations (powers, multiplication, addition, subtraction), you must follow the correct order of operations.
Worked Example: Complex Expression
Question: Evaluate , given , , and .
Solution:
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Write the expression:
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Substitute the values:
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Calculate powers first:
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Then multiplication:
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Finally addition and subtraction:
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Final answer:
Subtracting a negative number is the same as adding, so . This is because two negative signs together make a positive.
Common mistakes to avoid
Watch Out for These Common Errors:
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Forgetting brackets: Always put substituted values in brackets, especially negative numbers
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Order of operations: Remember to calculate powers before multiplication, and multiplication before addition/subtraction
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Negative number errors: Be extra careful when substituting or calculating with negative values
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Rushing the calculation: Take your time to work through each step methodically
Key Points to Remember:
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Substitution means replacing letters (variables) with numbers in an algebraic expression
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Always use brackets when substituting values to avoid errors
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Follow the four-step process: write the expression, substitute, evaluate, write your answer
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Respect the order of operations (BIDMAS/BODMAS): Brackets, Indices (powers), Division/Multiplication, Addition/Subtraction
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Non-linear expressions contain powers or roots and need extra care when evaluating
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When in doubt, work slowly and check each step of your calculation