Solve The Equation:$\[ 5x + \frac{1}{4} = \frac{61}{4} \\]

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 that involve fractions. We will use the equation $5x + \frac{1}{4} = \frac{61}{4}$ as an example to demonstrate the step-by-step process of solving linear equations with fractions.

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. In the equation $5x + \frac{1}{4} = \frac{61}{4}$, the highest power of x is 1, and the equation is in the form $ax + b = c$.

Understanding the Equation

Before we start solving the equation, let's take a closer look at it. The equation is $5x + \frac{1}{4} = \frac{61}{4}$. We can see that the equation involves a fraction, $\frac{1}{4}$, and a constant, $\frac{61}{4}$. The variable x is multiplied by 5, and then added to $\frac{1}{4}$.

Step 1: Subtract $\frac{1}{4}$ from Both Sides

To solve the equation, we need to isolate the variable x. The first step is to subtract $\frac{1}{4}$ from both sides of the equation. This will help us get rid of the fraction on the left-hand side of the equation.

{ 5x + \frac{1}{4} - \frac{1}{4} = \frac{61}{4} - \frac{1}{4} \} { 5x = \frac{60}{4} \}

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation is $\frac{60}{4}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

{ \frac{60}{4} = \frac{15}{1} \}

So, the equation becomes:

{ 5x = 15 \}

Step 3: Divide Both Sides by 5

Now that we have the equation in the form $5x = 15$, we can divide both sides by 5 to solve for x.

{ \frac{5x}{5} = \frac{15}{5} \} { x = 3 \}

Conclusion

In this article, we solved the linear equation $5x + \frac{1}{4} = \frac{61}{4}$ using the step-by-step process of subtracting $\frac{1}{4}$ from both sides, simplifying the right-hand side, and dividing both sides by 5. We hope that this article has provided a clear and concise explanation of how to solve linear equations with fractions.

Tips and Tricks

• When solving linear equations with fractions, it's essential to follow the order of operations (PEMDAS) to ensure that you perform the operations in the correct order.
• When subtracting fractions, make sure to subtract the numerators and keep the denominators the same.
• When simplifying fractions, look for common factors between the numerator and the denominator to simplify the fraction.

Common Mistakes to Avoid

• When solving linear equations with fractions, it's easy to get confused and make mistakes. Some common mistakes to avoid include:
• Forgetting to subtract the fraction from both sides of the equation.
• Not simplifying the fraction on the right-hand side.
• Dividing both sides by the wrong number.

Real-World Applications

Linear equations with fractions have many real-world applications, including:

• Finance: Linear equations with fractions can be used to calculate interest rates, investment returns, and other financial metrics.
• Science: Linear equations with fractions can be used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Linear equations with fractions can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations with fractions using the step-by-step process of subtracting fractions, simplifying the right-hand side, and dividing both sides by the coefficient of the variable. In this article, we will answer some frequently asked questions about solving linear equations with fractions.

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

A: The first step in solving a linear equation with fractions is to subtract the fraction from both sides of the equation. This will help you get rid of the fraction on the left-hand side of the equation.

Q: How do I simplify a fraction on the right-hand side of the equation?

A: To simplify a fraction on the right-hand side of the equation, look for common factors between the numerator and the denominator. If you find a common factor, divide both the numerator and the denominator by that factor to simplify the fraction.

Q: What if the fraction on the right-hand side of the equation is not simplified?

A: If the fraction on the right-hand side of the equation is not simplified, you can simplify it by finding the least common multiple (LCM) of the denominators and multiplying both the numerator and the denominator by that LCM.

Q: How do I divide both sides of the equation by a fraction?

A: To divide both sides of the equation by a fraction, multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Q: What if I get a negative value for the variable?

A: If you get a negative value for the variable, it means that the variable is negative. Make sure to check your work and ensure that the negative value is correct.

Q: Can I use a calculator to solve linear equations with fractions?

A: Yes, you can use a calculator to solve linear equations with fractions. However, make sure to check your work and ensure that the calculator is set to the correct mode (e.g., fraction mode).

Q: How do I check my work when solving linear equations with fractions?

A: To check your work when solving linear equations with fractions, plug the value of the variable back into the original equation and simplify. If the equation is true, then your work is correct.

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

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

• Forgetting to subtract the fraction from both sides of the equation.
• Not simplifying the fraction on the right-hand side.
• Dividing both sides by the wrong number.
• Not checking your work.

Q: Can I use linear equations with fractions to model real-world problems?

A: Yes, you can use linear equations with fractions to model real-world problems. Linear equations with fractions can be used to model population growth, chemical reactions, and other scientific phenomena.

Conclusion

In conclusion, solving linear equations with fractions is a crucial skill for students to master. By following the step-by-step process outlined in this article and answering the frequently asked questions, students can confidently solve linear equations with fractions and apply their knowledge to real-world problems.

Additional Resources

• Khan Academy: Solving Linear Equations with Fractions
• Mathway: Solving Linear Equations with Fractions
• Wolfram Alpha: Solving Linear Equations with Fractions

Practice Problems

• Solve the linear equation: $2x + \frac{1}{3} = \frac{5}{3}$
• Solve the linear equation: $x - \frac{2}{5} = \frac{3}{5}$
• Solve the linear equation: $3x + \frac{1}{2} = \frac{7}{2}$