Solve Equations using Multiplication and Division

Welcome to the world of algebraic problem-solving! In this article, titled “How to Solve Equations using Multiplication and Division,” we will explore the fundamental techniques that Grade 7 students need to solve equations involving multiplication and division.

Table of Contents:

• How to Solve Equations using Multiplication and Division
• Formula
• Solved Example
• Real-life Applications
• FAQs

How to Solve Equations using Multiplication and Division

Equations are powerful tools for representing relationships and finding unknown values, and mastering the skills to solve them is essential for success in mathematics.

By understanding the step-by-step processes and strategies involved in using multiplication and division, students will gain the confidence to tackle a variety of algebraic problems.

Understand the Equation:
Start by carefully reading and understanding the equation. Identify the variable (typically represented by a letter) and the values or expressions on both sides of the equation.

Simplify the Equation:
If there are any parentheses, brackets, or like terms, simplify the equation by applying the order of operations. Combine like terms and perform any necessary operations within parentheses or brackets.

Isolate the Variable:
The goal is to isolate the variable on one side of the equation. To do this, use inverse operations. If the variable is multiplied by a number, divide both sides of the equation by that number. If the variable is divided by a number, multiply both sides of the equation by that number.

Remember that whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality.

Simplify and Solve:
After isolating the variable, simplify the equation further if necessary. Then, solve for the variable by performing the indicated operation(s). Simplify both sides of the equation until the variable is determined.

Check the Solution:
To ensure the accuracy of the solution, substitute the found value back into the original equation. Verify that both sides of the equation are equal.

Formula

Multiplication Property of Equality:
The multiplication property of equality states that if both sides of an equation are multiplied by the same nonzero number, the equality remains unchanged. It can be represented as follows:
If a = b, then a * c = b * c, where “a,” “b,” and “c” represent any real numbers.

Division Property of Equality:
The division property of equality states that if both sides of an equation are divided by the same nonzero number, the equality remains unchanged. It can be represented as follows:
If a = b, then a / c = b / c, where “a,” “b,” and “c” represent any real numbers and c ≠ 0.

Inverse Operations:
Inverse operations are operations that undo each other. In the context of solving equations, multiplication and division are inverse operations. If a variable is multiplied by a number, you can divide both sides of the equation by that number to isolate the variable, and vice versa.

Cross-Multiplication:
Cross-multiplication is a technique used when solving equations involving fractions or ratios. It is based on the fact that the product of the means equals the product of the extremes in proportion. For example, in the equation a/b = c/d, you can cross-multiply by multiplying “a” with “d” and “b” with “c”.

Solved Examples

Formula 1: Multiplication Property of Equality
If a = b, then a * c = b * c, where “a,” “b,” and “c” represent any real numbers.

Example:
Solve the equation 2x = 10.

Solution:
We have the equation 2x = 10.

Using the multiplication property of equality, we can multiply both sides of the equation by 1/2 to isolate the variable x.

(2x) * (1/2) = 10 * (1/2)
x = 5

The solution is x = 5.

Formula 2: Division Property of Equality
If a = b, then a / c = b / c, where “a,” “b,” and “c” represent any real numbers and c ≠ 0.

Example:
Solve the equation 3y/4 = 9.

Solution:
We have the equation 3y/4 = 9.

Using the division property of equality, we can multiply both sides of the equation by 4/3 (the reciprocal of 3/4) to isolate the variable y.

(3y/4) * (4/3) = 9 * (4/3)
y = 12

The solution is y = 12.

Formula 3: Cross-Multiplication
In equations involving fractions or ratios, cross-multiplication can be used to solve for the unknown variable.

Example:
Solve the equation (2x + 3)/5 = 7.

Solution:
We have the equation (2x + 3)/5 = 7.
To eliminate the fraction, we can cross-multiply by multiplying 5 with 7 and (2x + 3).

5 * (2x + 3) = 7 * 5
10x + 15 = 35

Next, we can solve for x by isolating the variable.

10x = 35 – 15
10x = 20
x = 2

The solution is x = 2.

Formula 4: Inverse Operations
Inverse operations are operations that undo each other. In the context of solving equations, multiplication and division are inverse operations. If a variable is multiplied by a number, you can divide both sides of the equation by that number to isolate the variable, and vice versa.

Example:
Solve the equation 3x/2 – 4 = 7.

Solution:
We have the equation 3x/2 – 4 = 7.

To isolate the variable x, we can first add 4 to both sides of the equation.

3x/2 – 4 + 4 = 7 + 4
3x/2 = 11

Next, to eliminate the fraction, we can multiply both sides of the equation by 2/3 (the reciprocal of 3/2).

(3x/2) * (2/3) = 11 * (2/3)
x = 22/3

The solution is x = 22/3 or x = 7 1/3.

Real-life Applications

Shopping and Sales:

When shopping, understanding how to solve equations involving multiplication and division can help calculate discounts, sales prices, and final costs. For instance, determining the discounted price of an item after a percentage off or calculating the total cost of multiple items with different prices and quantities.

Recipe Scaling:

In cooking and baking, being able to solve equations involving multiplication and division is essential for scaling recipes. Adjusting ingredient quantities based on the desired serving size ensures accurate and consistent results. Solving equations helps determine the appropriate amounts of ingredients to maintain the same proportions.

Financial Planning:

Solving equations using multiplication and division is useful in financial planning, budgeting, and investing. It allows individuals to calculate interest rates, determine loan repayments, understand compound interest, and plan for future expenses or savings goals.

Engineering and Construction:

Equations involving multiplication and division are applied in engineering and construction fields. Architects, engineers, and builders use these skills to calculate measurements, scale drawings, determine material quantities, and solve construction-related problems.

Time and Distance Calculations:

Understanding how to solve equations using multiplication and division helps in calculating time and distance. For example, determining travel time based on a given speed and distance, or calculating rates of change in physical processes.

Sports and Athletics:

Equations involving multiplication and division are used in sports and athletics to analyze performance and track progress. Athletes and coaches can calculate average speeds, determine distances covered, or convert units of measurement for training and competition purposes.

FAQs

Grade 7 Science Worksheets

• Elements and Compounds
• Solar Energy
• Photosynthesis
• Electricity and Magnetism
• Law of conservation of energy
• Periodic table
• Properties of Matter
• Waves
• Energy Resources 
• Weather and Climate
• Immune, Circulatory and Digestive Systems
• Organs in Multi-cellular Organism
• Sedimentary, Igneous, and Metamorphic Rocks
• Structure of the Earth
• Law of Conservation of Mass
• Physical and Chemical Changes
• Scientific Method
• Human Digestive System
• Environmental Science
• Renewable and Non-renewable energy Resources
• Characteristics of Living Organisms
• Life Science
• Earth and Space Science
• Solar Eclipse
• Heat Technology
• Newton’s Laws of Motions
• Physical Science
• Tools, Measurement and SI Units
• Earth Atmosphere
• Interactions of Living things
• The Earth Ecosystem
• Organelles in Plant and Animal cells
• Layers of the Earth
• Cycles in Nature

Grade 7 Math Worksheets

• Fractions
• Linear equations word problems
• Statistics
• Properties of Parallel Line
• Finding slope from an equation
• Identifying Quadrilaterals
• Percent Change
• Properties of addition and multiplication
• Pythagorean Theorem
• Solving two step inequalities
• Symmetry
• Fractions to Decimals (New)
• Whole Number Exponents with Integer Bases (New)
• Adding and Subtracting Fractions (New)
• Integer Addition and Subtraction (New)
• Dividing Mixed Numbers (New)
• Basics of Coordinate Geometry (New)