NCERT Class 8 Maths: Linear Equation in One Variable - Exercise 2.6

Understanding Linear Equations with Variables on Both Sides (and Fractions)

In previous exercises, you learned to solve linear equations where the variable appeared on only one side, or where variables were on both sides but without fractions. Exercise 2.6 introduces equations where you might see fractions, or expressions that look complicated, but can be simplified into a standard linear equation form (ax + b = cx + d). The key idea remains the same: we want to isolate the variable on one side of the equation. When you have fractions on both sides of the equation, or an expression like (ax+b)/(cx+d) = k, the most common and effective technique is cross-multiplication. This method helps eliminate the denominators, transforming the fractional equation into a simpler linear equation without fractions. Remember, a linear equation means the highest power of the variable is always 1. Our goal is to manipulate the equation using basic algebraic operations (addition, subtraction, multiplication, division) while maintaining balance, until the variable stands alone.

Step-by-Step Method for Solving Complex Linear Equations

1. Step 1: Simplify Both Sides (if needed) — If there are any parentheses or like terms on either side of the equation, simplify them first. For equations with fractions, ensure each side is a single fraction or expression.
2. Step 2: Cross-Multiplication (for fractions) — If your equation is in the form of a fraction equal to another fraction, like a/b = c/d, perform cross-multiplication. This means multiplying the numerator of the left side by the denominator of the right side, and setting it equal to the product of the denominator of the left side and the numerator of the right side. So, a × d = b × c. If one side is not a fraction, treat it as k/1.
3. Step 3: Expand and Remove Parentheses — After cross-multiplication, you'll often have expressions with parentheses. Use the distributive property to expand and remove these parentheses. For example, 3(x + 2) becomes 3x + 6.
4. Step 4: Collect Like Terms — Gather all terms containing the variable (e.g., 'x' terms) on one side of the equation (usually the left side) and all constant terms (numbers without variables) on the other side (usually the right side). Remember to change the sign of a term when you move it from one side to the other.
5. Step 5: Isolate the Variable — Once you have the variable terms combined and constant terms combined (e.g., Ax = B), divide both sides of the equation by the coefficient of the variable (A) to find the value of the variable. Remember, whatever operation you do on one side, you must do on the other to keep the equation balanced.
6. Step 6: Verify Your Solution (Optional but Recommended) — Substitute the value you found for the variable back into the original equation. If both sides of the equation are equal, your solution is correct. This step helps catch any calculation errors.

Worked Examples from NCERT Ex 2.6

• Example 1: Solve (8x - 3) / (3x) = 2 Step 1: The equation is already in the form a/b = c/1. Step 2: Perform cross-multiplication. (8x - 3) × 1 = 2 × (3x) 8x - 3 = 6x Step 3: Collect variable terms on one side and constants on the other. 8x - 6x = 3 2x = 3 Step 4: Isolate the variable. x = 3/2 Final Answer: x = 3/2
• Example 2: Solve (9x) / (7 - 6x) = 15 Step 1: The equation is (9x) / (7 - 6x) = 15/1. Step 2: Perform cross-multiplication. 9x × 1 = 15 × (7 - 6x) 9x = 105 - 90x Step 3: Collect variable terms on one side. 9x + 90x = 105 99x = 105 Step 4: Isolate the variable. x = 105 / 99 x = 35 / 33 (by dividing both numerator and denominator by 3) Final Answer: x = 35/33
• Example 3: Solve (z) / (z + 15) = 4 / 9 Step 1: The equation is already in the form a/b = c/d. Step 2: Perform cross-multiplication. z × 9 = 4 × (z + 15) 9z = 4z + 60 Step 3: Collect variable terms on one side. 9z - 4z = 60 5z = 60 Step 4: Isolate the variable. z = 60 / 5 z = 12 Final Answer: z = 12

Exam Tip: Avoiding Common Mistakes in Ex 2.6

When dealing with linear equations involving fractions and variables on both sides, students often make a few common errors. First, be extremely careful with sign changes when moving terms from one side of the equation to the other. A + becomes a -, and a - becomes a +. Second, during cross-multiplication, remember to multiply the entire expression in the denominator or numerator. If there's an expression like (7 - 6x), the number outside the bracket (e.g., 15) must be multiplied by both terms inside: 15 × 7 - 15 × 6x. Forgetting to distribute properly is a frequent mistake. Always double-check your calculations, especially when dealing with negative numbers. Finally, always verify your answer by substituting it back into the original equation to ensure both sides are equal. This simple check can save you from losing marks!

Practice Questions with Solutions

• Q: Solve: (x + 1) / (2x + 3) = 3 / 8 A: Step 1: Perform cross-multiplication. 8 × (x + 1) = 3 × (2x + 3) Step 2: Distribute on both sides. 8x + 8 = 6x + 9 Step 3: Gather variable terms on one side and constants on the other. 8x - 6x = 9 - 8 2x = 1 Step 4: Isolate the variable. x = 1/2 Final answer: x = 1/2
• Q: Solve: 5x / (x + 5) = 5 A: Step 1: Rewrite the equation as 5x / (x + 5) = 5/1. Step 2: Perform cross-multiplication. 5x × 1 = 5 × (x + 5) 5x = 5x + 25 Step 3: Gather variable terms on one side. 5x - 5x = 25 0 = 25 Step 4: Analyse the result. Since 0 is not equal to 25, there is no solution to this equation. Final answer: No solution
• Q: Solve: (7y + 4) / (y + 2) = -4 / 3 A: Step 1: Perform cross-multiplication. 3 × (7y + 4) = -4 × (y + 2) Step 2: Distribute on both sides. 21y + 12 = -4y - 8 Step 3: Gather variable terms on one side and constants on the other. 21y + 4y = -8 - 12 25y = -20 Step 4: Isolate the variable. y = -20 / 25 y = -4 / 5 Final answer: y = -4/5
• Q: Solve: (2 - 5x) / (1 - 2x) = 1 / 3 A: Step 1: Perform cross-multiplication. 3 × (2 - 5x) = 1 × (1 - 2x) Step 2: Distribute on both sides. 6 - 15x = 1 - 2x Step 3: Gather variable terms on one side and constants on the other. 6 - 1 = -2x + 15x 5 = 13x Step 4: Isolate the variable. x = 5 / 13 Final answer: x = 5/13

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