Some examples of expressions we have so far worked with are | Class 8 Mathematics Notes

2.2 Solving Equations having the Variable on both Sides

An equation is an equality between two expressions. In many cases, the right-hand side (RHS) of an equation is a number, but sometimes both sides contain expressions with variables. For example, the equation 2x - 3 = x + 2 has variables on both sides. Here, the LHS is 2x - 3 and the RHS is x + 2.

To solve such equations, we perform algebraic operations to isolate the variable on one side. The key principle is to maintain equality by performing the same operation on both sides.

Example 1: Solve 2x - 3 = x + 2.

Step 1: Add 3 to both sides to eliminate the constant term on the LHS: 2x - 3 + 3 = x + 2 + 3 which simplifies to: 2x = x + 5

Step 2: Subtract x from both sides to bring variable terms to one side: 2x - x = x + 5 - x which simplifies to: x = 5

Thus, the solution is x = 5.

Example 2: Solve 5x + 7/2 = (3/2)x - 14.

Step 1: Multiply both sides by 2 (the LCM of denominators) to clear fractions: 2 × (5x + 7/2) = 2 × ((3/2)x - 14) which simplifies to: 10x + 7 = 3x - 28

Step 2: Transpose 3x to the LHS by subtracting 3x from both sides: 10x - 3x + 7 = -28 which simplifies to: 7x + 7 = -28

Step 3: Subtract 7 from both sides: 7x = -28 - 7 which simplifies to: 7x = -35

Step 4: Divide both sides by 7: x = -5

Thus, the solution is x = -5.

This section emphasizes that variables can be transposed from one side to the other by performing inverse operations, and fractions can be cleared by multiplying both sides by the LCM of denominators. These techniques simplify the solving process.

📊 Diagram: Figure on page 2 illustrating balance and operations on equations.

🧪 Activity: No specific activity; includes worked examples demonstrating solving equations with variables on both sides.

🔗 Connection: Prepares for the next section on reducing equations to simpler forms by opening brackets and combining like terms.