Solve The Equation For { X $} : : : { 3x + 9 = 8x - 6 \}

Mar 9, 2025 by ADMIN 57 views

Solving Linear Equations: A Step-by-Step Guide to Isolating the Variable

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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 a specific type of linear equation, where the variable is isolated on one side of the equation. We will use the equation $3x + 9 = 8x - 6$ as an example to demonstrate the step-by-step process of solving for the variable.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it represents. The equation $3x + 9 = 8x - 6$ is a linear equation in one variable, where the variable is represented by $x$ . The equation states that the sum of $3x$ and $9$ is equal to the sum of $8x$ and $-6$ .

Key Components of the Equation

Coefficient of x: The coefficient of $x$ is the number that is multiplied by the variable. In this equation, the coefficient of $x$ is $3$ and $8$ .
Constant Term: The constant term is the number that is not multiplied by the variable. In this equation, the constant term is $9$ and $-6$ .

Solving the Equation

To solve the equation, we need to isolate the variable $x$ on one side of the equation. We can do this by using the following steps:

Step 1: Add or Subtract the Same Value to Both Sides

The first step is to add or subtract the same value to both sides of the equation to eliminate the constant term. In this case, we can add $6$ to both sides of the equation to get:

3x + 9 + 6 = 8x - 6 + 6

This simplifies to:

3x + 15 = 8x

Step 2: Subtract the Same Value from Both Sides

The next step is to subtract the same value from both sides of the equation to isolate the variable. In this case, we can subtract $3x$ from both sides of the equation to get:

3x + 15 - 3x = 8x - 3x

This simplifies to:

15 = 5x

Step 3: Divide Both Sides by the Coefficient of x

The final step is to divide both sides of the equation by the coefficient of $x$ to solve for the variable. In this case, we can divide both sides of the equation by $5$ to get:

\frac{15}{5} = \frac{5x}{5}

This simplifies to:

3 = x

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation by isolating the variable. We used the equation $3x + 9 = 8x - 6$ as an example to illustrate the process. By following the steps outlined above, we were able to solve for the variable $x$ and arrive at the solution $x = 3$ .

Frequently Asked Questions

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation by using the steps outlined above.

Q: What is the coefficient of x?

A: The coefficient of $x$ is the number that is multiplied by the variable.

Q: What is the constant term?

A: The constant term is the number that is not multiplied by the variable.

Tips and Tricks

Tip 1: Use the Order of Operations

When solving a linear equation, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the calculations correctly.

Tip 2: Check Your Work

After solving a linear equation, it's crucial to check your work by plugging the solution back into the original equation to ensure that it is true.

Tip 3: Practice, Practice, Practice

The more you practice solving linear equations, the more comfortable you will become with the process.

Resources

Online Resources

Khan Academy: Linear Equations
Mathway: Linear Equations
Wolfram Alpha: Linear Equations

Textbooks

"Algebra and Trigonometry" by Michael Sullivan
"College Algebra" by James Stewart

Videos

"Solving Linear Equations" by Khan Academy
"Linear Equations" by Math Antics

Conclusion

Solving linear equations is a fundamental skill that is essential for success in mathematics and other fields. By following the steps outlined above and practicing regularly, you can become proficient in solving linear equations and tackle more complex problems with confidence.

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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 address some of the most frequently asked questions about linear equations and provide detailed answers to help you better understand the concept.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation by using the following steps:

Add or subtract the same value to both sides of the equation to eliminate the constant term.
Subtract the same value from both sides of the equation to isolate the variable.
Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the coefficient of x?

A: The coefficient of x is the number that is multiplied by the variable. In the equation ax + b = c, the coefficient of x is a.

Q: What is the constant term?

A: The constant term is the number that is not multiplied by the variable. In the equation ax + b = c, the constant term is b and c.

Q: How do I check my work when solving a linear equation?

A: To check your work, plug the solution back into the original equation to ensure that it is true. If the solution satisfies the equation, then it is correct.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not following the order of operations (PEMDAS)
Not isolating the variable on one side of the equation
Not checking your work
Not using the correct signs when adding or subtracting

Q: How do I graph a linear equation?

A: To graph a linear equation, use the slope-intercept form of the equation (y = mx + b), where m is the slope and b is the y-intercept. Plot the y-intercept on the graph, and then use the slope to find the other points on the line.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, use the following methods:

Substitution method: Substitute the expression for one variable into the other equation.
Elimination method: Add or subtract the equations to eliminate one variable.
Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Tips and Tricks

Tip 1: Use the Order of Operations

When solving a linear equation, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the calculations correctly.

Tip 2: Check Your Work

After solving a linear equation, it's crucial to check your work by plugging the solution back into the original equation to ensure that it is true.

Tip 3: Practice, Practice, Practice

The more you practice solving linear equations, the more comfortable you will become with the process.