Exercise 2.3 — Variables on Both Sides

Equations with Variables on Both Sides

In earlier exercises, you solved equations where the right-hand side (RHS) was simply a number — for example, 2x + 3 = 9. In Exercise 2.3, the challenge steps up: both sides of the equation contain variable terms. This is a critical skill tested in Class 8 board exams across CBSE, Telangana, and Andhra Pradesh syllabi. The core idea remains the same — use transposition to collect all variable terms on one side and all constants on the other.

The Key Strategy: Transposition

Transposition means moving a term from one side of the equation to the other, changing its sign in the process. When both sides have variable terms, follow this two-step approach:

• Collect variable terms on the LHS — transpose all variable terms from the RHS to the left side.
• Collect constants on the RHS — transpose all numbers from the LHS to the right side.
• Simplify both sides — combine like terms, then divide both sides by the coefficient of the variable.
• When brackets are involved — expand using the distributive law first, then transpose.

Worked Examples from Exercise 2.3

Simple variable-on-both-sides (Q1): Consider 7x − 5 = 2x. Transpose 2x to the LHS and −5 to the RHS to get 5x = 5, giving x = 1.

Both sides have variables and constants (Q3): For 7p − 3 = 3p + 8, transpose 3p to the LHS and −3 to the RHS: 4p = 11, so p = 11/4. Notice the answer is a fraction — this is perfectly valid and common in board exams.

Brackets on one side (Q7): The equation 3x + 4 = 5(x − 2) requires expanding the bracket first.

Brackets on both sides (Q11): For 15(x − 1) + 4(x + 3) = 2(7 + x), expand all brackets, group variable terms on the LHS and constants on the RHS:

Multiple brackets, complex grouping (Q12): 3(5z − 7) + 2(9z − 11) = 4(8z − 7) − 111 involves expanding three brackets and careful collection of like terms, giving z = −96. This type of question frequently appears in Telangana and AP state board question papers.

Common Mistakes to Avoid

• Sign errors during transposition — when you move a term across the equals sign, its sign must flip. Moving −5 to the RHS gives +5, not −5.
• Incorrect bracket expansion — multiply every term inside the bracket by the factor outside. For 5(x − 2), both x and −2 must be multiplied by 5.
• Not simplifying like terms before transposing — in multi-bracket problems (like Q11–Q14), expand all brackets first and combine like terms on each side before moving terms across.
• Rejecting fractional answers — solutions like 11/4 or −9/5 are completely correct. Do not assume the answer must be a whole number.
• Forgetting to verify — substitute your answer back into the original equation to confirm LHS = RHS. This habit can save marks in board exams.

What This Exercise Prepares You For

Mastering equations with variables on both sides directly supports several topics across Class 8, 9, and 10. In Class 8, this skill is applied in Exercise 2.4 word problems, where real-life situations are translated into exactly these types of equations. As you move into Class 9 and 10, the same transposition technique is used in solving systems of linear equations and forms the foundation for working with quadratic equations. Students preparing for CBSE, Telangana Board (TS), or Andhra Pradesh Board (AP) exams will find this exercise among the most frequently tested in the algebra section.