Algebra
Linear Equations
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to solve one-step linear equations.
- ✓By the end of this lesson students will be able to solve two-step linear equations.
- ✓By the end of this lesson students will be able to solve linear equations with unknowns on both sides.
- ✓By the end of this lesson students will be able to solve linear equations involving brackets.
- ✓By the end of this lesson students will be able to check their solutions by substitution.
Key concepts
An equation that involves an unknown (usually represented by a letter like 'x') raised to the power of one. It does not contain terms like x², x³, or square roots of x. The aim is to find the value of the unknown that makes the equation true.
The process of finding the value(s) of the unknown variable(s) that satisfy the equation. This means finding the value(s) that make both sides of the equation equal.
The fundamental principle of solving equations. Whatever operation you perform on one side of the equals sign, you must perform the exact same operation on the other side to maintain the equality. Think of it like a set of scales – to keep them balanced, if you add weight to one side, you must add the same weight to the other.
Operations that undo each other. For example, addition is the inverse of subtraction, and multiplication is the inverse of division. We use inverse operations to isolate the unknown variable.
The process of combining terms that have the same variable part (e.g., 3x and 5x) or are constant numbers (e.g., 7 and -2). This simplifies the equation before solving.
The process of multiplying each term inside the bracket by the term outside the bracket. For example, 3(x + 2) expands to 3x + 6. This is often the first step when solving equations with brackets.
Key facts to remember
- 1An equation contains an equals sign (=), indicating that the expression on the left has the same value as the expression on the right.
- 2The goal of solving a linear equation is to isolate the unknown variable (e.g., x) on one side of the equation.
- 3To maintain balance, any operation performed on one side of the equation must also be performed on the other side.
- 4Use inverse operations to 'undo' operations and move terms across the equals sign (e.g., add to undo subtract, divide to undo multiply).
- 5When solving, deal with addition/subtraction first, then multiplication/division (reverse order of operations, BIDMAS/BODMAS).
- 6If an equation contains brackets, expand them first.
- 7Collect all terms involving the unknown variable on one side and all constant terms on the other side.
- 8Always check your solution by substituting the value back into the original equation to ensure both sides are equal.
Worked examples
Example 1
Solve 3x - 7 = 11
Answer
x = 6
This is a standard two-step equation. Always deal with addition/subtraction first, then multiplication/division.
Example 2
Solve 5x + 3 = 2x + 15
Answer
x = 4
It's often easier to move the smaller 'x' term to avoid negative coefficients, but not essential.
Example 3
Solve 4(x - 2) = 2x + 10
Answer
x = 9
Always expand brackets as the first step when they are present in an equation.
Common mistakes
- ✗Forgetting to apply an operation to *both* sides of the equation, leading to an unbalanced equation.
- ✗Incorrectly applying inverse operations, for example, subtracting 3 when the equation has '+3', or multiplying by 2 when the equation has '/2'.
- ✗Making errors with negative numbers, especially when collecting terms or performing inverse operations.
- ✗Incorrectly expanding brackets, such as 3(x + 2) becoming 3x + 2 instead of 3x + 6.
- ✗Not collecting like terms correctly, for instance, combining '3x' and '5' to get '8x' or '8'.
Exam tips
- ★Show every step of your working clearly. This allows the examiner to follow your logic and awards method marks even if you make a calculation error.
- ★Always check your final answer by substituting it back into the original equation. If both sides are equal, your answer is correct.
- ★Work systematically, dealing with brackets first, then collecting terms, then isolating the variable.
- ★If you find yourself with a negative unknown (e.g., -x = 5), remember to multiply or divide both sides by -1 to find the positive value of the unknown (x = -5).
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