Linear Equations in One Variable: Class 8 Maths NCERT Guide

What is a Linear Equation in One Variable?

Let's break down the name. An expression is a mathematical phrase like x + 5 or 2y - 3. It doesn't have an 'equals' sign. When we set two expressions equal to each other, we get an equation, like x + 5 = 10. The letter x is a variable, which is just a placeholder for a number we don't know yet. A linear equation means the highest power of the variable is just 1 (so we see x, not x² or x³). And finally, 'in one variable' means there's only one type of unknown in the whole equation (like only x, not x and y together). So, a linear equation in one variable is an equation with one unknown letter, where that letter's power is 1. The main goal is always to find the one special value for the variable that makes the left side of the equation exactly equal to the right side.

How to Solve Linear Equations: The Balancing Method

1. Step 1: Understand the Goal — The goal is to find the value of the variable (like x). To do this, you need to get the variable all by itself on one side of the equals sign (=).
2. Step 2: Transposition - Move Constants — Move all the plain numbers (constants) to one side of the equation (usually the right-hand side or RHS). When a number crosses the '=' sign, it changes its sign. A + becomes a -, and a - becomes a +.
3. Step 3: Transposition - Move Variables — If the variable appears on both sides, move all terms with the variable to the other side (usually the left-hand side or LHS). Just like with constants, their signs will change when they move.
4. Step 4: Simplify and Isolate the Variable — Combine the terms on both sides. You will be left with something like ax = b. To find x, divide both sides by the number attached to x (which is a). So, x = b/a.

Worked Examples of Solving Linear Equations

• Example 1: Solve the equation 2x - 3 = 7 Step 1: We want to get x alone. First, let's move the constant -3 to the right side. 2x = 7 + 3 Step 2: Simplify the right side. 2x = 10 Step 3: The variable x is multiplied by 2. To isolate x, we do the opposite: divide both sides by 2. x = 10 / 2 Final Answer: x = 5. We can check this: 2(5) - 3 = 10 - 3 = 7. The equation is balanced!
• Example 2: Solve the equation 5x + 9 = 5 + 3x Step 1: Move the variable term 3x from the right side to the left side. It becomes -3x. 5x - 3x + 9 = 5 Step 2: Move the constant term +9 from the left side to the right side. It becomes -9. 5x - 3x = 5 - 9 Step 3: Simplify both sides. 2x = -4 Step 4: Isolate x by dividing both sides by 2. x = -4 / 2 Final Answer: x = -2. Checking our work: 5(-2) + 9 = -10 + 9 = -1. And 5 + 3(-2) = 5 - 6 = -1. Both sides are equal!

Exam Tip: Avoid Common Mistakes

The most frequent error students make is with signs during transposition. Always remember: when a term moves to the other side of the '=' sign, its operation inverts. Addition (+) becomes subtraction (-), and multiplication (*) becomes division (/). Double-check your signs at every step! Another common mistake is in equations like (x+3)/2 = 5. Remember to move the /2 first by multiplying the other side by 2, before you move the +3. Always deal with division/multiplication affecting the whole side first, then handle the addition/subtraction inside.

Practice Questions with Solutions

• Q: Solve for y: 8y = 24 A: Step 1: The variable y is multiplied by 8. To isolate y, we need to perform the opposite operation, which is division. Step 2: Divide both sides of the equation by 8. 8y / 8 = 24 / 8 Step 3: Simplify the equation. y = 3 Final answer: y = 3.
• Q: Find the solution of: 3x - 10 = 5 A: Step 1: Move the constant term -10 from the left side to the right side. Its sign changes to +. 3x = 5 + 10 Step 2: Simplify the right side. 3x = 15 Step 3: Isolate x by dividing both sides by 3. x = 15 / 3 Final answer: x = 5.
• Q: Solve the equation: z/4 + 7 = 12 A: Step 1: First, move the constant +7 to the right side. It becomes -7. z/4 = 12 - 7 Step 2: Simplify the right side. z/4 = 5 Step 3: The variable z is divided by 4. To isolate z, multiply both sides by 4. z = 5 * 4 Final answer: z = 20.
• Q: Solve for p: 9p + 1 = 4p + 26 A: Step 1: Move the variable term 4p to the left side. It becomes -4p. 9p - 4p + 1 = 26 Step 2: Move the constant term +1 to the right side. It becomes -1. 9p - 4p = 26 - 1 Step 3: Simplify both sides. 5p = 25 Step 4: Isolate p by dividing both sides by 5. p = 25 / 5 Final answer: p = 5.

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