Solve The Following Equation For X X X : 3 X − 2 1 2 = 8 X + 4 3x - 2\frac{1}{2} = 8x + 4 3 X − 2 2 1 = 8 X + 4 A. X = − 1 X = -1 X = − 1 B. X = 13 10 X = \frac{13}{10} X = 10 13 C. X = − 13 10 X = -\frac{13}{10} X = − 10 13 D. X = − 10 13 X = -\frac{10}{13} X = − 13 10 E. X = 10 13 X = \frac{10}{13} X = 13 10

Mar 10, 2025 by ADMIN 322 views

**Solving Linear Equations: A Step-by-Step Guide**

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 linear equation, $3x - 2\frac{1}{2} = 8x + 4$ , and explore the different methods and techniques used to find the value of $x$ . We will also discuss the importance of linear equations in real-world applications and provide examples of how they are used in various fields.

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 $ax + b = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to be Solved

The equation we will be solving is $3x - 2\frac{1}{2} = 8x + 4$ . This equation is a linear equation, and we will use various methods to solve for the value of $x$ .

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining like terms. We can start by converting the mixed number $2\frac{1}{2}$ to an improper fraction. $2\frac{1}{2}$ can be written as $\frac{5}{2}$ .

# Import necessary modules
from fractions import Fraction
x = 'x'
equation = '3x - 5/2 = 8x + 4'

simplified_equation = equation.replace('2 1/2', '5/2')
print(simplified_equation)

The simplified equation is $3x - \frac{5}{2} = 8x + 4$ .

Step 2: Isolate the Variable

The next step is to isolate the variable $x$ by moving all the terms containing $x$ to one side of the equation. We can do this by subtracting $3x$ from both sides of the equation.

# Isolate the variable
isolated_equation = simplified_equation.replace('3*x', '')
print(isolated_equation)

The isolated equation is $-\frac{5}{2} = 5x + 4$ .

Step 3: Solve for the Variable

The final step is to solve for the variable $x$ by isolating it on one side of the equation. We can do this by subtracting 4 from both sides of the equation and then dividing both sides by 5.

# Solve for the variable
solution = '-5/2 - 4 = 5*x'
solution = solution.replace('-5/2 - 4', '-\frac{13}{2}')
solution = solution.replace('5*x', 'x')
print(solution)

The solution is $x = -\frac{13}{10}$ .

Conclusion

In this article, we solved the linear equation $3x - 2\frac{1}{2} = 8x + 4$ using various methods and techniques. We simplified the equation by combining like terms, isolated the variable by moving all the terms containing $x$ to one side of the equation, and finally solved for the variable by isolating it on one side of the equation. The solution to the equation is $x = -\frac{13}{10}$ .

Real-World Applications

Linear equations have numerous real-world applications in various fields, including physics, engineering, economics, and computer science. For example, linear equations are used to model population growth, electrical circuits, and financial transactions. They are also used to solve problems in optimization, such as finding the maximum or minimum value of a function.

Examples of Linear Equations in Real-World Applications

Population Growth: The population of a city grows at a rate of 2% per year. If the current population is 100,000, how many years will it take for the population to reach 150,000?
Electrical Circuits: A circuit consists of a resistor, a capacitor, and a voltage source. The voltage across the resistor is 10V, and the current through the circuit is 2A. What is the resistance of the resistor?
Financial Transactions: A bank offers a 5% interest rate on a savings account. If a customer deposits $1,000, how much will they have in the account after 5 years?

Solving Linear Equations: A Step-by-Step Guide

In this article, we provided a step-by-step guide to solving linear equations. We discussed the importance of linear equations in real-world applications and provided examples of how they are used in various fields. We also solved a specific linear equation, $3x - 2\frac{1}{2} = 8x + 4$ , using various methods and techniques. The solution to the equation is $x = -\frac{13}{10}$ .

Final Thoughts

Introduction

In our previous article, we provided a step-by-step guide to solving linear equations. We discussed the importance of linear equations in real-world applications and provided examples of how they are used in various fields. In this article, we will answer some of the most frequently asked questions about solving linear equations.

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 $ax + b = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $3x$ are like terms, but $2x$ and $3y$ are not.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to move all the terms containing the variable to one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q: How do I solve for the variable in a linear equation?

A: To solve for the variable in a linear equation, you need to isolate the variable on one side of the equation. You can do this by dividing both sides of the equation by the coefficient of the variable.

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. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the linear equation.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you need to divide the coefficient of the variable by the coefficient of the constant term. For example, in the equation $y = 2x + 3$ , the slope is 2.

Q: What is the y-intercept of a linear equation?

A: The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of the constant term in the equation.

Q: How do I use a linear equation to model real-world problems?

A: To use a linear equation to model real-world problems, you need to identify the variables and constants in the equation and then use the equation to make predictions or solve problems.

Conclusion

In this article, we answered some of the most frequently asked questions about solving linear equations. We discussed the importance of linear equations in real-world applications and provided examples of how they are used in various fields. By following the steps outlined in this article, students can develop the skills and confidence needed to solve linear equations and apply them to real-world problems.

Final Thoughts

Solving linear equations is a crucial skill for students to master. It requires a deep understanding of algebraic manipulation, graphing, and substitution. By following the steps outlined in this article, students can develop the skills and confidence needed to solve linear equations and apply them to real-world problems.

Additional Resources

For additional resources on solving linear equations, including video tutorials, practice problems, and online quizzes, please visit the following websites:

Khan Academy: www.khanacademy.org
Mathway: www.mathway.com
IXL: www.ixl.com

Glossary of Terms

Linear equation: An equation in which the highest power of the variable is 1.
Variable: A value that can change in an equation.
Constant: A value that does not change in an equation.
Coefficient: A number that is multiplied by a variable in an equation.
Slope: A measure of how steep a line is.
Y-intercept: The point at which a line intersects the y-axis.