Algebra
Linear Equations & Rearranging Formulae
Year 10 · Year 11
- ✓By the end of this lesson students will be able to solve linear equations with one unknown, including those with brackets and fractions.
- ✓By the end of this lesson students will be able to rearrange formulae to change the subject.
- ✓By the end of this lesson students will be able to apply the principles of inverse operations to solve equations and rearrange formulae.
- ✓By the end of this lesson students will be able to understand and correctly use the term 'subject' in the context of a formula.
Key concepts
A linear equation is an algebraic equation where the highest power of the unknown variable (e.g., x, y, t) is 1. It can be written in the general form ax + b = c, where a, b, and c are constants and a ≠ 0. The graph of a linear equation is a straight line.
Solving a linear equation means finding the specific value of the unknown variable that makes the equation true. This is achieved by isolating the variable on one side of the equals sign, using inverse operations to 'undo' the operations performed on the variable.
Rearranging a formula, also known as changing the subject, involves manipulating the formula to express one particular variable in terms of the others. The variable that is isolated on one side of the equals sign is called the 'subject' of the formula. The same inverse operations used for solving equations are applied.
Key facts to remember
- 1To solve an equation or rearrange a formula, perform the same operation on both sides of the equals sign.
- 2Use inverse operations: addition undoes subtraction, multiplication undoes division.
- 3When expanding brackets, multiply every term inside the bracket by the term outside.
- 4When moving a term from one side of the equation to the other, change its sign.
- 5If the variable you want to make the subject appears in more than one term, factorise it out.
- 6The 'subject' of a formula is the variable that is isolated on one side of the equals sign.
Worked examples
Example 1
Solve the equation: 4x - 5 = 19
Answer
x = 6
Always perform the same operation on both sides to keep the equation balanced.
Example 2
Solve the equation: 3(2y + 1) = 5y - 4
Answer
y = -7
Be careful with negative numbers when moving terms across the equals sign.
Example 3
Make x the subject of the formula: ax + b = cx + d
Answer
x = (d - b) / (a - c)
When the subject appears in multiple terms, factorising is a crucial step.
Common mistakes
- ✗Forgetting to apply an operation to both sides of the equation, leading to an unbalanced equation.
- ✗Incorrectly expanding brackets, for example, writing 2(x+3) as 2x+3 instead of 2x+6.
- ✗Making sign errors when moving terms across the equals sign (e.g., changing x - 5 to x = 10 - 5 instead of x = 10 + 5).
- ✗Not factorising the desired subject when it appears in multiple terms, preventing its isolation.
- ✗Incorrectly dealing with fractions, such as multiplying only part of an equation by the denominator.
Exam tips
- ★Show all your working steps clearly. Method marks are often awarded even if the final answer is incorrect.
- ★Check your solution by substituting your answer back into the original equation to ensure it holds true.
- ★Be meticulous with negative numbers and signs throughout your calculations; these are common sources of error.
- ★Practise rearranging formulae with different letters and structures to become comfortable with the process, rather than just memorising steps for specific examples.
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