Solve The Equation, Keeping The Value For \[$x\$\] As An Improper Fraction.$\[ \frac{2}{3}x = -\frac{1}{2}x + 5 \\]1. Isolate The Variable By Adding \[$\frac{1}{2}x\$\] To Both Sides:$\[ \frac{2}{3}x + \frac{1}{2}x =

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, keeping the value of the variable as an improper fraction. We will break down the solution into manageable steps, making it easy to understand and follow.

The Equation

The given equation is:

$\frac{2}{3}x = -\frac{1}{2}x + 5$

Our goal is to isolate the variable $x$ and express it as an improper fraction.

Step 1: Add $\frac{1}{2}x$ to Both Sides

To isolate the variable, we need to get all the terms containing $x$ on one side of the equation. We can do this by adding $\frac{1}{2}x$ to both sides of the equation:

$\frac{2}{3}x + \frac{1}{2}x = -\frac{1}{2}x + \frac{1}{2}x + 5$

This simplifies to:

$\frac{2}{3}x + \frac{1}{2}x = 5$

Step 2: Find a Common Denominator

To add the fractions on the left-hand side, we need to find a common denominator. The least common multiple (LCM) of 3 and 2 is 6. We can rewrite the fractions with a common denominator of 6:

$\frac{4}{6}x + \frac{3}{6}x = 5$

This simplifies to:

$\frac{7}{6}x = 5$

Step 3: Multiply Both Sides by the Reciprocal

To isolate $x$, we need to get rid of the fraction on the left-hand side. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{7}{6}$, which is $\frac{6}{7}$:

$\frac{6}{7} \cdot \frac{7}{6}x = \frac{6}{7} \cdot 5$

This simplifies to:

$x = \frac{30}{7}$

Conclusion

In this article, we solved a linear equation by isolating the variable $x$ and expressing it as an improper fraction. We followed a step-by-step approach, adding $\frac{1}{2}x$ to both sides, finding a common denominator, and multiplying both sides by the reciprocal. By mastering these skills, students can confidently solve linear equations and apply them to real-world problems.

Tips and Variations

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to simplify fractions whenever possible.
• In some cases, it may be necessary to multiply both sides of the equation by a fraction to eliminate the variable. Be sure to multiply both sides by the reciprocal of the fraction.
• Linear equations can be used to model real-world problems, such as calculating the cost of goods, determining the amount of time it takes to complete a task, or finding the area of a shape.

Practice Problems

Try solving the following linear equations:

1. $\frac{3}{4}x = 2x - 1$
2. $\frac{2}{5}x + 1 = \frac{3}{5}x$
3. $\frac{1}{2}x - 2 = \frac{1}{3}x + 1$

Introduction

In our previous article, we explored the step-by-step process of solving linear equations, keeping the value of the variable as an improper fraction. In this article, we will address some common questions and concerns that students may have when solving linear equations.

Q: What is a linear equation?

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

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

• The highest power of the variable is 1.
• The equation can be written in the form ax = b.
• The equation does not contain any exponents or roots.

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:

• Linear equation: 2x = 5
• Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, follow these steps:

1. Add or subtract fractions to get rid of the fractions on the left-hand side.
2. Multiply both sides of the equation by the reciprocal of the fraction to eliminate the variable.
3. Simplify the equation and solve for the variable.

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

Q: How do I simplify fractions?

A: To simplify a fraction, follow these steps:

1. Find the greatest common divisor (GCD) of the numerator and denominator.
2. Divide both the numerator and denominator by the GCD.
3. Simplify the resulting fraction.

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 simplifying fractions.
• Not isolating the variable on one side of the equation.
• Not checking the solution to ensure it is correct.

Conclusion

In this article, we addressed some common questions and concerns that students may have when solving linear equations. By following the steps outlined in this article and avoiding common mistakes, students can confidently solve linear equations and apply them to real-world problems.

Practice Problems

Try solving the following linear equations:

1. $\frac{3}{4}x = 2x - 1$
2. $\frac{2}{5}x + 1 = \frac{3}{5}x$
3. $\frac{1}{2}x - 2 = \frac{1}{3}x + 1$

Remember to follow the steps outlined in this article and to simplify fractions whenever possible.