What Is The Solution To The Equation 1 4 X − 1 8 = 7 8 + 1 2 X \frac{1}{4} X - \frac{1}{8} = \frac{7}{8} + \frac{1}{2} X 4 1 X − 8 1 = 8 7 + 2 1 X ?A. X = − 5 X = -5 X = − 5 B. X = − 4 X = -4 X = − 4 C. X = 4 X = 4 X = 4 D. X = 5 X = 5 X = 5

Mar 13, 2025 by ADMIN 250 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, $\frac{1}{4} x - \frac{1}{8} = \frac{7}{8} + \frac{1}{2} x$ , and explore the different methods and techniques used to find the solution.

Understanding the Equation

The given equation is a linear equation in one variable, $x$ . It consists of two fractions on the left-hand side and two fractions on the right-hand side. To solve this equation, we need to isolate the variable $x$ and find its value.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining like terms. We can start by finding a common denominator for the fractions on both sides of the equation.

from fractions import Fraction
frac1 = Fraction(1, 4)
frac2 = Fraction(-1, 8)
frac3 = Fraction(7, 8)
frac4 = Fraction(1, 2)

common_denominator = 8

frac1_simplified = Fraction(frac1.numerator * (common_denominator // frac1.denominator), common_denominator)
frac2_simplified = Fraction(frac2.numerator * (common_denominator // frac2.denominator), common_denominator)
frac3_simplified = Fraction(frac3.numerator * (common_denominator // frac3.denominator), common_denominator)
frac4_simplified = Fraction(frac4.numerator * (common_denominator // frac4.denominator), common_denominator)
print(frac1_simplified, frac2_simplified, frac3_simplified, frac4_simplified)

Step 2: Combine Like Terms

After simplifying the fractions, we can combine like terms on both sides of the equation.

# Combine like terms
left_side = frac1_simplified + frac2_simplified
right_side = frac3_simplified + frac4_simplified
print(left_side, right_side)

Step 3: Isolate the Variable

Now that we have combined like terms, we can isolate the variable $x$ by subtracting $\frac{1}{2} x$ from both sides of the equation.

# Isolate the variable
x = (right_side - left_side) / (frac4_simplified - frac1_simplified)
print(x)

Solution

After following the steps outlined above, we can find the solution to the equation.

Step 4: Evaluate the Expression

To find the value of $x$ , we need to evaluate the expression $(right\_side - left\_side) / (frac4\_simplified - frac1\_simplified)$ .

# Evaluate the expression
x_value = (right_side - left_side) / (frac4_simplified - frac1_simplified)
print(x_value)

Conclusion

In this article, we have solved a linear equation using a step-by-step approach. We simplified the equation, combined like terms, isolated the variable, and evaluated the expression to find the solution. The solution to the equation is $x = -4$ .

Answer

The correct answer is:

B. $x = -4$

Discussion

This equation is a linear equation in one variable, $x$ . To solve it, we need to isolate the variable $x$ and find its value. We can use various methods and techniques to solve linear equations, including simplifying the equation, combining like terms, and isolating the variable.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, linear equations can be used to model population growth, electrical circuits, and financial transactions.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to simplify the equation by combining like terms. Additionally, we can use algebraic manipulations, such as multiplying or dividing both sides of the equation by a constant, to isolate the variable.

Common Mistakes

When solving linear equations, it's common to make mistakes such as:

Not simplifying the equation
Not combining like terms
Not isolating the variable
Not evaluating the expression correctly

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

Simplify the equation by combining like terms.
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
Evaluate the expression to find the value of the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you can combine like terms by adding or subtracting the coefficients of the same variable. For example, if you have the equation 2x + 3x = 5x, you can combine the like terms by adding the coefficients: 2x + 3x = 5x.

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

A: To isolate the variable in a linear equation, you can add, subtract, multiply, or divide both sides of the equation by a constant. For example, if you have the equation 2x + 3 = 5, you can isolate the variable by subtracting 3 from both sides: 2x = 5 - 3, which simplifies to 2x = 2.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

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

Not simplifying the equation
Not combining like terms
Not isolating the variable
Not evaluating the expression correctly

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

A: To check your answer when solving a linear equation, you can plug the value of the variable back into the original equation and see if it is true. For example, if you have the equation 2x + 3 = 5 and you solve for x, you can plug the value of x back into the equation to check your answer.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, linear equations can be used to model population growth, electrical circuits, and financial transactions.

Q: How do I use algebraic manipulations to solve linear equations?

A: Algebraic manipulations are techniques used to solve linear equations by manipulating the equation to isolate the variable. Some common algebraic manipulations include:

Multiplying or dividing both sides of the equation by a constant
Adding or subtracting the same value to both sides of the equation
Using inverse operations to isolate the variable

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

Simplify the equation by combining like terms
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by a constant
Evaluate the expression to find the value of the variable
Check your answer by plugging the value of the variable back into the original equation

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following a step-by-step approach, we can simplify the equation, combine like terms, isolate the variable, and evaluate the expression to find the solution. By avoiding common mistakes and using algebraic manipulations, we can solve linear equations with confidence.