Key Learning Objectives
By the end of this lesson and revision module, Year 7 students will be able to:
- Understand how to use inverse operations to balance and solve one-step and two-step linear equations.
- Isolate the unknown variable algebraically while keeping the equation balanced on both sides.
- Substitute calculated values back into the original equation to verify accuracy.
Topic Summary & Essential Notes
In Year 7 Algebra, an equation is a mathematical statement showing that two expressions are equal in value. To solve an equation means to find the exact numerical value of the unknown variable (often represented by letters such as x, y, or n). The golden rule of algebra is balance: whatever operation you apply to one side of the equals sign, you must apply to the exact opposite side. We isolate the variable by systematically applying inverse operations (addition is the inverse of subtraction, and multiplication is the inverse of division).
Key Rules, Formulas & Definitions
Inverse Operations Rule
+ ↔ − (Addition cancels Subtraction) and × ↔ ÷ (Multiplication cancels Division)
Golden Rule of Balance
Perform identical operations on BOTH the Left-Hand Side (LHS) and Right-Hand Side (RHS)
General Two-Step Structure
For ax + b = c, first subtract b from both sides (ax = c − b), then divide by a (x = (c − b) / a)
2 Step-by-Step Worked Examples
Study these model solutions to understand how examiners award method and accuracy marks:
Worked Example 1
Worked Example 1: Solving a Two-Step Linear Equation
Question: Solve the equation: 3x + 7 = 22
Step-by-Step Solution:
- Step 1: Identify and eliminate the constant term (+7). Apply the inverse operation (−7) to both sides:
3x + 7 − 7 = 22 − 73x = 15 - Step 2: Eliminate the coefficient of x (3). Since 3x means 3 multiplied by x, divide both sides by 3:
3x ÷ 3 = 15 ÷ 3x = 5 - Step 3: Verification. Substitute x = 5 back into the original equation: 3(5) + 7 = 15 + 7 = 22. The solution is confirmed.
Final Answer: x = 5
Senior Examiner Insight: Always write each step on a new line with aligned equals signs. Examiners award 1 method mark for applying inverse operations even if arithmetic slips occur.
Worked Example 2
Worked Example 2: Solving an Equation with Division and Subtraction
Question: Solve the equation: \(\frac{y}{4} - 3 = 9\)
Step-by-Step Solution:
- Step 1: Eliminate the constant (−3). Add 3 to both sides:
y/4 − 3 + 3 = 9 + 3y/4 = 12 - Step 2: Eliminate the division by 4. Multiply both sides by 4:
(y/4) × 4 = 12 × 4y = 48 - Step 3: Check: 48 ÷ 4 = 12, and 12 − 3 = 9. Correct.
Final Answer: y = 48
Senior Examiner Insight: Do not multiply by 4 before adding 3 unless you multiply every single term (including the −3) by 4. Adding 3 first is the cleanest, least error-prone method.
Common Pitfalls & Examiner Warnings
Avoid these common mistakes frequently identified by examiners in Key Stage 3 assessments:
Common Mistake: Forgetting to apply operations to BOTH sides
A common mistake is subtracting 7 from the left side but forgetting to subtract it from the right side, completely unbalancing the equation.
Common Mistake: Sign errors with negative numbers
When solving 2x − 8 = 10, students often mistakenly subtract 8 instead of adding 8.
Common Mistake: Skipping the verification step
Failing to substitute the solution back into the original equation means easily avoidable calculation mistakes go unnoticed.
Interactive Self-Check Quiz (3 Questions)
Test your understanding before checking the full worked answer and examiner mark scheme:
Question 1 of 3
What is the value of x in the equation 4x − 5 = 19?
Click to Reveal Answer & Mark Scheme
Correct Answer: B) x = 6
Detailed Explanation: Add 5 to both sides: 4x = 19 + 5 = 24. Divide both sides by 4: x = 24 ÷ 4 = 6.
Mark Scheme & Scoring Points: 1 mark for method: 4x = 24; 1 mark for correct final answer: x = 6.
Question 2 of 3
Solve: 5(2y + 1) = 35. What is the value of y?
Click to Reveal Answer & Mark Scheme
Correct Answer: A) y = 3
Detailed Explanation: Method 1: Divide both sides by 5 first: 2y + 1 = 7. Subtract 1: 2y = 6. Divide by 2: y = 3.
Method 2: Expand brackets: 10y + 5 = 35. Subtract 5: 10y = 30. Divide by 10: y = 3.
Method 2: Expand brackets: 10y + 5 = 35. Subtract 5: 10y = 30. Divide by 10: y = 3.
Mark Scheme & Scoring Points: 1 mark for expanding brackets or dividing by 5; 1 mark for correct solution y = 3.
Question 3 of 3
A student solves 3n + 12 = 6 and writes n = 6. What error did they make?
Click to Reveal Answer & Mark Scheme
Correct Answer: B) They calculated 6 − 12 = 6 instead of −6, forgetting negative integers.
Detailed Explanation: Subtracting 12 from 6 gives 6 − 12 = −6. Then 3n = −6, so n = −6 ÷ 3 = −2. The student incorrectly treated 6 − 12 as 12 − 6 = 6.
Mark Scheme & Scoring Points: Award 1 mark for identifying the negative number subtraction error.
Created by Fiaraz Iqbal
Former Headteacher & Senior Science / Maths Examiner
With over 15 years of leadership and classroom experience, Fiaraz creates high-yield revision resources aligned with National Curriculum standards to help Key Stage 3 and GCSE students secure top grades.
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