Step-by-Step Guide to Solving Equations with Multiplication or Division
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 \( x/3 = 5 \):
- Multiply both sides by 3: \( x/3 \times 3 = 5 \times 3 \)
- So, \( x = 15 \)
Example 2: Solving by Division
Solve \( 7x = 28 \):
- Divide both sides by 7: \( 7x/7 = 28/7 \)
- So, \( x = 4 \)
Example 3: Negative Numbers
Solve \( x/(-2) = 6 \):
- Multiply both sides by -2: \( x/(-2) \times (-2) = 6 \times (-2) \)
- So, \( x = -12 \)
Example 4: Fractional Coefficient
Solve \( 0.5y = 7 \):
- 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.
FAQs on Solving Equations Using Multiplication and Division Made Easy
1. How can you solve an equation using multiplication and division?
To solve equations using multiplication and division, apply the inverse operation to isolate the variable. If a variable is divided, multiply both sides by the divisor; if multiplied, divide both sides by the coefficient. This maintains the equation's balance and reveals the variable's value.
2. What is the operation of multiplication and division?
Multiplication and division are inverse operations in algebra. Multiplication finds the product of two numbers; division finds how many times one number is contained in another. In equation solving, they are used to isolate the variable by performing the opposite operation on both sides to maintain equality.
3. What is the order of operations for multiplication and division?
In the order of operations (PEMDAS/BODMAS), multiplication and division have equal precedence and are performed from left to right. When solving equations, we reverse this order, addressing addition/subtraction first, then multiplication/division in reverse to isolate the variable.
4. How to solve an equation with multiple operations?
Solving equations with multiple operations involves applying the order of operations (PEMDAS/BODMAS) in reverse. Begin by undoing addition/subtraction, then multiplication/division, working step-by-step to isolate the variable, ensuring you perform the same operation on both sides of the equation.
5. What is the property of equality in multiplication/division?
The property of equality states that if you perform any operation (multiplication or division) on one side of an equation, you must perform the same operation on the other side to maintain the equation's balance. This ensures the solution remains accurate.
6. How do you reverse an operation?
To reverse an operation, use its inverse operation. The inverse of addition is subtraction, the inverse of multiplication is division, and vice versa. This is the fundamental principle for solving equations. For example, to undo multiplying by 5, divide by 5.
7. What if x is divided/multiplied?
If 'x' is divided by a number, multiply both sides of the equation by that number to isolate 'x'. If 'x' is multiplied by a number, divide both sides by that number to isolate 'x'. Remember to always perform the same operation on both sides to maintain balance.
8. What’s the formula for solving multiplication/division equations?
There isn't one single formula, but rather a method: If x/a = b, then x = a * b. If a * x = b, then x = b / a. These are based on the inverse operation principle. The key is understanding the inverse relationship between multiplication and division.
9. How do I approach a multi-step or negative value equation?
For multi-step equations, apply the order of operations (PEMDAS/BODMAS) in reverse. For negative values, treat the negative sign as part of the number. Remember to maintain balance by performing the same operation on both sides of the equation throughout the process. Carefully track negative signs during calculations.
10. Where can I download more practice materials?
Downloadable practice worksheets and further resources are available on the Vedantu website. These materials provide additional practice problems to help you master solving equations using multiplication and division, covering different difficulty levels and problem types.
11. What are some real-life examples of multiplication/division equations?
Real-life applications of multiplication and division in equation solving include calculating unit prices (cost per item), determining proportions in recipes, or calculating speed/distance/time problems. Many situations require finding an unknown value by 'undoing' multiplications and divisions.
12. What if I get the wrong answer?
If you get the wrong answer, double-check your calculations. Common mistakes include errors in applying inverse operations, forgetting to operate on both sides of the equation equally, or mismanaging negative signs. Always substitute your answer back into the original equation to verify its accuracy. Review the steps and identify where the error occurred.
13. Why isn’t my method working?
If your method isn't working, carefully review the order of operations. Make sure you're applying inverse operations correctly and consistently to both sides of the equation. Errors often stem from incorrect application of the inverse operation or order of operations. Check for calculation errors, especially sign errors with negative numbers.
