Solving Equations by Multiplying and Dividing Both Sides

Solving equations by using different operations such as multiplication and division is a fundamental skill in algebra, vital for school and competitive exams. This technique helps students learn how to isolate variables and find solutions efficiently. Mastering these basics sets the foundation for advanced topics and real-life problem-solving.

Understanding Solving Equations Using Multiplication and Division

An equation is a statement with an equal sign where two expressions are set as equal. To solve for the unknown variable, we often need to undo mathematical operations. Multiplication and division are powerful tools for "undoing" opposite operations or balancing both sides of an equation. If a variable is multiplied by a number, you reverse it by dividing; if divided, reverse with multiplication.

For example, in the equation \( x/4 = 5 \), since the variable is divided by 4, you use multiplication to isolate \( x \).

Formulae and Properties for Solving Equations

Here are the common forms and how to solve them:

• If \( x/a = b \), then \( x = a \times b \)
• If \( a \times x = b \), then \( x = b/a \)

These use the property of equality: whatever operation you do to one side of the equation, you must do to the other. This keeps the equation balanced.

Real-life application: If 24 candies are shared equally among 6 friends, the equation is \( x \times 6 = 24 \). To find x (candies per friend), divide 24 by 6.

Step-by-Step Worked Examples

Example 1: Solving by Multiplication

Solve \( x/3 = 5 \):

1. Multiply both sides by 3: \( x/3 \times 3 = 5 \times 3 \)
2. So, \( x = 15 \)

Example 2: Solving by Division

Solve \( 7x = 28 \):

1. Divide both sides by 7: \( 7x/7 = 28/7 \)
2. So, \( x = 4 \)

Example 3: Negative Numbers

Solve \( x/(-2) = 6 \):

1. Multiply both sides by -2: \( x/(-2) \times (-2) = 6 \times (-2) \)
2. So, \( x = -12 \)

Example 4: Fractional Coefficient

Solve \( 0.5y = 7 \):

1. Divide both sides by 0.5: \( y = 7 / 0.5 = 14 \)

Practice Problems

• Solve \( x/4 = 9 \)
• Solve \( 8y = 56 \)
• Solve \( z/(-3) = -12 \)
• Solve \( 2x = 1.6 \)
• Solve \( y/6 = 2.5 \)

Common Mistakes to Avoid

• Not applying the same operation to both sides of the equation.
• Dividing by zero (which is not allowed in mathematics).
• Confusing when to multiply and when to divide while isolating the variable.
• Missing negative signs or mishandling them during multiplication/division.
• Forgetting to check the solution by substituting back into the original equation.

Real-World Applications

Solving equations with multiplication and division is useful in everyday life—for instance, calculating unit prices in shopping, converting currencies, distributing materials equally in a recipe, or figuring out speed and distance in travel. In science, such equations help in chemistry (balancing reaction equations) and physics (calculating force, work, or pressure).

At Vedantu, we show students how to apply these skills to daily scenarios and exam questions, making learning practical and engaging. For more practice on similar concepts, see linear equations in one variable or review multiplication facts at tables 2 to 20.

In this topic, you learned how solving equations using multiplication and division works, step-by-step examples, formulas, and common mistakes to watch out for. These skills are essential for success in exams and future math topics. Practice regularly and use resources on Vedantu to strengthen your understanding of solving equations with different operations.