Linear Equations — REET Maths Notes
Solving linear equations in one variable is a skill that comes up regularly in the Algebra section of the REET maths paper. This page explains the transposition method, the steps for solving an equation, and how to turn word problems into equations, all with worked examples, so that age, number and quantity-type word problems become easy to solve.
- 0 questions in REET 2024
- Class 7–8 level
- 9 min read
How this topic is asked in REET
The Algebra section of the REET maths syllabus includes basic building blocks like equations. No direct question from this specific sub-topic showed up in the REET 2024 paper, so this page is built on standard syllabus facts and the NCERT Class 7-8 maths textbook. Typically, a question asks you to solve a simple equation in one variable, or to convert a word problem into an equation and solve it. Word problems on ages, sums and differences of numbers, and the perimeter of a rectangle are repeated the most, so practising these specifically is important.
Defining a linear equation
An equation in which the highest power of the variable is 1 is called a linear equation — for example, x − 2 = 7 or 3x + 5 = 20. Solving it means finding the value of the variable at which both sides of the equation (LHS and RHS) become equal.
The key difference between an equation and an expression is that an equation always has an 'equals' (=) sign, while an expression doesn't — for example, '3x + 5' is an expression, but '3x + 5 = 20' is an equation. Solving an equation gives you a fixed value of the variable, whereas the value of an expression can change with every different value of the variable.
Rules for solving equations
| Rule | Explanation | Example |
|---|---|---|
| Same-operation rule | Adding, subtracting, multiplying or dividing both sides of an equation by the same number does not change the equation | 6x = 12 → dividing both sides by 6 gives x = 2 |
| Transposition rule | Moving a term from one side of the equation to the other flips its sign (+ becomes −, − becomes +) | x − 2 = 7 → x = 7 + 2 = 9 |
Transposition is really just a shortcut version of the same-operation rule — instead of adding/subtracting the same number on both sides, you directly flip the term's sign and move it to the other side
Steps for solving
- Step 1
- Bring all variable terms to one side of the equation and all constant (numerical) terms to the other side, using transposition
- Step 2
- Combine like terms on both sides to simplify the equation
- Step 3
- Divide both sides by the coefficient of the variable to find its value
- Step 4
- Substitute the value you found back into the original equation to verify it
- Caution
- If the equation contains brackets, expand them (multiply out) first, and only then begin the process of transposition
A worked example
Example: solve 5x − 3 = 3x + 7. Bringing variable terms to the left and constant terms to the right gives: 5x − 3x = 7 + 3, that is, 2x = 10, so x = 5. Verification: the left side gives 5(5) − 3 = 22, the right side gives 3(5) + 7 = 22 — both sides are equal, so x = 5 is the correct solution. This verification step is also useful in the exam, because it lets you quickly rule out wrong answers among the options.
Word problem examples
| Problem | Equation | Solution |
|---|---|---|
| The sum of two numbers is 95. If one number is 15 more than the other, find the numbers. | Let the smaller number = a, the larger number = a + 15; so a + (a + 15) = 95 | 2a + 15 = 95 ⟹ 2a = 80 ⟹ a = 40; the numbers are 40 and 55 |
| The length of a rectangle is 5 cm more than its width, and its perimeter is 34 cm. Find the width. | Let the width = a, the length = a + 5; perimeter = 2 × (length + width) = 34 | 2(a + a + 5) = 34 ⟹ 2(2a + 5) = 34 ⟹ 4a + 10 = 34 ⟹ 4a = 24 ⟹ a = 6 cm |
How to teach this to children
Classroom activity
Draw a physical balance scale to show that both sides of an equation are in balance; adding a number to one pan means you must add the exact same number to the other pan to keep the balance — repeat this across several different equations.
Teaching aid
Use an unknown-weight box for the variable and weight blocks for constants to solve simple equations like 'x + 3 = 8' in a concrete, hands-on way, then move gradually to harder equations.
Common misconception
Students often forget to flip the sign of a term during transposition (for example, writing x = 7 − 2 from x − 2 = 7); fix this by repeating 'change sides, change signs' regularly and having students verify every solution afterward.
Assessment idea
Have students build word problems based on real situations — their own age, pocket money, or a sibling's age — turn these into equations, solve them, then share solutions in class and check each other's answers.
A trick worth remembering
'Change sides, change signs' — whenever a term crosses over to the other side of the equals sign during transposition, its addition/subtraction sign flips; this is the single most useful rule in REET equations. When solving a word problem, always start by calling the unknown quantity x and writing the equation, then solve it, and finally check your answer against the original question.
Practice questions
Question 1
The solution of the equation x + 5 = 12 is—
- Ax = 5
- Bx = 7
- Cx = 17
- Dx = 8
Explanation
By transposition, x = 12 − 5 = 7.
Question 2
The solution of the equation 4x = 20 is—
- Ax = 4
- Bx = 5
- Cx = 16
- Dx = 24
Explanation
Dividing both sides by 4 gives x = 20 ÷ 4 = 5.
Question 3
The solution of the equation 3x − 7 = 8 is—
- Ax = 3
- Bx = 5
- Cx = 1
- Dx = 15
Explanation
By transposition, 3x = 8 + 7 = 15, so x = 15 ÷ 3 = 5.
Question 4
The solution of the equation 2x + 3 = x + 9 is—
- Ax = 6
- Bx = 3
- Cx = −6
- Dx = 12
Explanation
Bringing variable terms to one side and constants to the other: 2x − x = 9 − 3, so x = 6. Check: LHS = 2(6)+3 = 15, RHS = 6+9 = 15.
Question 5
The solution of the equation 5(x − 2) = 20 is—
- Ax = 4
- Bx = 6
- Cx = 2
- Dx = 10
Explanation
5x − 10 = 20 ⟹ 5x = 30 ⟹ x = 6.
Question 6
Adding 7 to three times a number gives 34. That number is—
- A7
- B9
- C11
- D27
Explanation
3x + 7 = 34 ⟹ 3x = 27 ⟹ x = 9.
Question 7
The sum of two numbers is 60 and their difference is 10. The larger number is—
- A25
- B30
- C35
- D40
Explanation
Let the larger number be a and the smaller be b: a + b = 60, a − b = 10; adding both gives 2a = 70, so a = 35.
Question 8
Ravi's age is 4 times his son's age. The sum of their ages is 40 years. The son's age is—
- A8 वर्ष
- B10 वर्ष
- C32 वर्ष
- D6 वर्ष
Explanation
Let the son's age be x, so Ravi's age is 4x; x + 4x = 40 ⟹ 5x = 40 ⟹ x = 8.
Question 9
The solution of the equation (x/3) + 2 = 7 is—
- Ax = 5
- Bx = 10
- Cx = 15
- Dx = 21
Explanation
x/3 = 7 − 2 = 5, so x = 5 × 3 = 15.
Question 10
A Class 7 teacher is teaching students to solve the equation 'x + 3 = 8' using the balance scale model. What do students understand best from this method?
- Aसमीकरण के दोनों पक्षों में समान संक्रिया करने पर संतुलन (बराबरी) बना रहता है
- Bसमीकरण को हल करने की कोई आवश्यकता नहीं
- Cचर का मान हमेशा शून्य होता है
- Dसमीकरण केवल गुणा से हल होते हैं
Explanation
The balance-scale model visually shows that performing the same operation on both sides of an equation keeps it in balance — this is the core idea behind solving a linear equation.
Frequently asked questions
How are linear equations asked about in REET?+
Usually as solving a simple equation in one variable, or converting a word problem into an equation and solving it.
What is the transposition method?+
It's a way of moving a term of the equation from one side to the other by flipping its sign (positive to negative, or vice versa), without changing the value of the equation.
How do you verify the solution of a linear equation?+
Substitute the value you found back into the variable's place in the original equation and check whether both sides (LHS and RHS) come out equal.
What is the maximum power of the variable in a linear equation with one variable?+
In a linear equation with one variable, the power of the variable is always 1; if the power is 2 or higher, the equation becomes quadratic or higher-order, not linear.
What is the difference between an equation and an expression?+
An equation always has an 'equals' (=) sign and gives one fixed solution, while an expression has no equals sign and its value keeps changing with every value of the variable.
Read next
Factorisation
Factorising expressions — the previous topic in the REET maths sequence
Ratio and Proportion
Ratio and proportion — the next topic in the REET maths sequence
REET Syllabus
The complete official syllabus for REET Level 1 and Level 2, in one place
REET 2024 Solved Paper
All solved REET 2024 questions, organised subject-wise
Change Log
- 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked