Common Linear Equations and Their Solutions

Oct 13, 2025 by miodeya.com 104 views

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear equations of the form ax = b, where a and b are constants. We will learn how to write the solution to each equation as a fraction and as a decimal.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax = b, where a and b are constants. For example, 2x = 3 is a linear equation, where a = 2 and b = 3.

Solving Linear Equations

To solve a linear equation, we need to isolate the variable (x) on one side of the equation. We can do this by performing inverse operations on both sides of the equation.

Step 1: Divide Both Sides by the Coefficient of x

The first step in solving a linear equation is to divide both sides of the equation by the coefficient of x. In the equation 2x = 3, the coefficient of x is 2. To divide both sides by 2, we can multiply both sides by the reciprocal of 2, which is 1/2.

2x = 3
x = 3 / 2

Step 2: Simplify the Fraction

The fraction 3/2 can be simplified by dividing the numerator (3) by the denominator (2). This gives us a decimal value of 1.5.

x = 3 / 2 = 1.5

Step 3: Write the Solution as a Decimal

Now that we have simplified the fraction, we can write the solution as a decimal. In this case, the solution is x = 1.5.

Example 2: Solving the Equation 4x = 12

Let's solve the equation 4x = 12 using the same steps.

Step 1: Divide Both Sides by the Coefficient of x

The coefficient of x in the equation 4x = 12 is 4. To divide both sides by 4, we can multiply both sides by the reciprocal of 4, which is 1/4.

4x = 12
x = 12 / 4

Step 2: Simplify the Fraction

The fraction 12/4 can be simplified by dividing the numerator (12) by the denominator (4). This gives us a decimal value of 3.

x = 12 / 4 = 3

Step 3: Write the Solution as a Decimal

Now that we have simplified the fraction, we can write the solution as a decimal. In this case, the solution is x = 3.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, we can solve linear equations of the form ax = b and write the solution as a fraction and as a decimal. Remember to divide both sides of the equation by the coefficient of x and simplify the fraction to get the solution.

Common Linear Equations and Their Solutions

Here are some common linear equations and their solutions:

Equation	Solution (Fraction)	Solution (Decimal)
2x = 3	3/2	1.5
4x = 12	12/4	3
3x = 9	9/3	3
5x = 15	15/5	3
2x = 6	6/2	3

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

Always divide both sides of the equation by the coefficient of x.
Simplify the fraction by dividing the numerator by the denominator.
Write the solution as a decimal by dividing the numerator by the denominator.
Use a calculator to check your answers.

Practice Problems

Here are some practice problems to help you practice solving linear equations:

Solve the equation 2x = 6.
Solve the equation 3x = 9.
Solve the equation 5x = 15.
Solve the equation 2x = 12.
Solve the equation 4x = 16.

Answer Key

Here are the answers to the practice problems:

x = 3
x = 3
x = 3
x = 6
x = 4

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about linear equations, along with their answers.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax = b, where a and b are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (x) on one side of the equation. You can do this by performing inverse operations on both sides of the equation.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to divide both sides of the equation by the coefficient of x.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator by the denominator.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a number as a ratio of two integers, while a decimal is a way of expressing a number as a sum of powers of 10.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you need to divide the numerator by the denominator.

Q: What is the coefficient of x?

A: The coefficient of x is the number that is multiplied by x in the equation.

Q: How do I find the coefficient of x?

A: To find the coefficient of x, you need to look at the equation and identify the number that is multiplied by x.

Q: What is the reciprocal of a number?

A: The reciprocal of a number is 1 divided by that number.

Q: How do I use the reciprocal to solve a linear equation?

A: To use the reciprocal to solve a linear equation, you need to multiply both sides of the equation by the reciprocal of the coefficient of x.

Q: What is the inverse operation of addition?

A: The inverse operation of addition is subtraction.

Q: What is the inverse operation of multiplication?

A: The inverse operation of multiplication is division.

Q: How do I use inverse operations to solve a linear equation?

A: To use inverse operations to solve a linear equation, you need to perform the inverse operation of the operation that is being performed on the variable (x).

Q: What is the solution to a linear equation?

A: The solution to a linear equation is the value of the variable (x) that makes the equation true.

Q: How do I write the solution to a linear equation as a fraction and as a decimal?

A: To write the solution to a linear equation as a fraction and as a decimal, you need to simplify the fraction and then convert it to a decimal.

Q: What are some common linear equations and their solutions?

A: Here are some common linear equations and their solutions:

Equation	Solution (Fraction)	Solution (Decimal)
2x = 3	3/2	1.5
4x = 12	12/4	3
3x = 9	9/3	3
5x = 15	15/5	3
2x = 6	6/2	3

Q: How do I practice solving linear equations?

A: To practice solving linear equations, you can try solving the following equations:

2x = 6
3x = 9
5x = 15
2x = 12
4x = 16

Q: What are some tips and tricks for solving linear equations?

A: Here are some tips and tricks for solving linear equations:

Always divide both sides of the equation by the coefficient of x.
Simplify the fraction by dividing the numerator by the denominator.
Write the solution as a decimal by dividing the numerator by the denominator.
Use a calculator to check your answers.