Strand 4 — Algebra
Rearranging Formulae
1st Year · 2nd Year · 3rd Year (Junior Cert)
- ✓By the end of this lesson students will be able to identify the subject of a formula.
- ✓By the end of this lesson students will be able to use inverse operations to rearrange formulae.
- ✓By the end of this lesson students will be able to make a specified variable the subject of a formula.
- ✓By the end of this lesson students will be able to rearrange formulae involving multiple steps and operations.
- ✓By the end of this lesson students will be able to apply rearranging formulae to solve problems.
Key concepts
A formula is a mathematical rule or relationship that shows how different quantities (represented by variables) are related to each other. For example, the formula for the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width.
The subject of a formula is the variable that is isolated on one side of the equals sign, usually on the left. It is expressed in terms of the other variables. For example, in the formula A = lw, 'A' is the subject.
Inverse operations are operations that undo each other. To rearrange a formula, we use inverse operations to move terms around and isolate the desired variable. * Addition (+) and Subtraction (-) are inverse operations. * Multiplication (×) and Division (÷) are inverse operations. * Squaring (²) and taking the square root (√) are inverse operations.
Key facts to remember
- 1The goal of rearranging a formula is to isolate the desired variable on one side of the equals sign.
- 2Whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side.
- 3Use inverse operations to 'undo' operations performed on the variable you want to isolate.
- 4Work backwards through the order of operations (e.g., deal with addition/subtraction before multiplication/division, unless the variable is grouped in a bracket).
- 5The subject of the formula is usually written on the left-hand side for clarity.
- 6Treat other variables in the formula as if they were known numbers.
Worked examples
Example 1
Make 'a' the subject of the formula P = a + b + c.
Answer
a = P - b - c
Remember to perform the same operation on both sides of the equals sign to maintain balance.
Example 2
Make 'w' the subject of the formula A = lw.
Answer
w = A / l
Treat other variables as if they were numbers when performing operations.
Example 3
Make 'l' the subject of the formula P = 2(l + w).
Answer
l = P / 2 - w
Work 'backwards' through the order of operations (e.g., deal with multiplication/division before addition/subtraction if the variable is inside a bracket).
Common mistakes
- ✗Forgetting to apply an operation to *both* sides of the equation.
- ✗Incorrectly applying inverse operations (e.g., adding when you should subtract, or multiplying when you should divide).
- ✗Making sign errors when moving terms across the equals sign (e.g., a positive term becomes negative when moved).
- ✗Not dealing with coefficients correctly (e.g., adding a coefficient instead of dividing by it).
- ✗Incorrectly expanding brackets or performing operations inside brackets before dealing with terms outside.
Exam tips
- ★Clearly show each step of your working. This helps you track your progress and allows for partial marks if you make a small error.
- ★Identify the variable you need to make the subject before you start manipulating the formula.
- ★Check your final rearranged formula by substituting simple numbers for the variables and seeing if the original and rearranged formulae hold true.
- ★Practise a variety of examples, including those with fractions, squares, and square roots, to become confident with different types of formulae.
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