Not distributing the fraction correctly

Oct 13, 2025 by leolavanderia.com.br 8 views

$Not distributing the fraction correctly$

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 using the distributive property. We will break down the solution into manageable steps, making it easier for readers to understand and follow along.

The Equation

The equation we will be solving is $\frac{1}{4}(16 + 12x) = 28$ . This equation involves a fraction and a variable, making it a bit more challenging to solve. However, with the right approach and steps, we can easily find the value of the variable.

Step 1: Distributing the Fraction

The first step in solving this equation is to distribute the fraction $\frac{1}{4}$ to the terms inside the parentheses. This means we need to multiply the fraction by each term inside the parentheses.

Distributing the Fraction to the Constant Term

To distribute the fraction to the constant term, we multiply $\frac{1}{4}$ by 16.

$\frac{1}{4} \times 16 = 4$

So, the first term in the equivalent equation is 4.

Distributing the Fraction to the Variable Term

Next, we need to distribute the fraction to the variable term, which is $12x$ . To do this, we multiply $\frac{1}{4}$ by $12x$ .

$\frac{1}{4} \times 12x = 3x$

So, the second term in the equivalent equation is $3x$ .

The Equivalent Equation

Now that we have distributed the fraction to both terms, we can write the equivalent equation.

$\frac{1}{4} \times 16 + \frac{1}{4} \times 12x = 28$

Simplifying this equation, we get:

$4 + 3x = 28$

Step 2: Isolating the Variable

The next step is to isolate the variable $x$ . To do this, we need to get rid of the constant term on the left-hand side of the equation.

Subtracting the Constant Term

To isolate the variable, we subtract 4 from both sides of the equation.

$4 + 3x - 4 = 28 - 4$

This simplifies to:

$3x = 24$

Step 3: Solving for the Variable

Now that we have isolated the variable, we can solve for $x$ . To do this, we need to get rid of the coefficient of $x$ , which is 3.

Dividing Both Sides by the Coefficient

To solve for $x$ , we divide both sides of the equation by 3.

$\frac{3x}{3} = \frac{24}{3}$

This simplifies to:

$x = 8$

Conclusion

In this article, we solved the linear equation $\frac{1}{4}(16 + 12x) = 28$ using the distributive property. We broke down the solution into manageable steps, making it easier for readers to understand and follow along. By distributing the fraction to the terms inside the parentheses, we were able to simplify the equation and isolate the variable. Finally, we solved for the variable by getting rid of the coefficient and dividing both sides of the equation by the coefficient. With this step-by-step guide, readers should be able to solve similar linear equations with ease.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. Here are a few:

Not distributing the fraction correctly: Make sure to distribute the fraction to all terms inside the parentheses.
Not isolating the variable correctly: Make sure to get rid of the constant term on the left-hand side of the equation before isolating the variable.
Not solving for the variable correctly: Make sure to get rid of the coefficient of the variable before solving for it.

Practice Problems

To practice solving linear equations, try the following problems:

$\frac{1}{2}(24 + 16x) = 36$
$\frac{3}{4}(12 + 20x) = 45$
$\frac{2}{3}(18 + 12x) = 24$

Real-World Applications

Linear equations have many real-world applications. Here are a few examples:

Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Conclusion

Introduction

In our previous article, we solved a linear equation using the distributive property. However, we know that solving linear equations can be a challenging task, especially for students who are new to the concept. In this article, we will provide a Q&A guide to help students understand and solve linear equations.

Q: What is a linear equation?

A: A linear equation 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 is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac. This property is used to simplify expressions and solve equations.

Q: How do I distribute the fraction in a linear equation?

A: To distribute the fraction in a linear equation, you need to multiply the fraction by each term inside the parentheses. For example, if you have the equation $\frac{1}{4}(16 + 12x) = 28$ , you would multiply $\frac{1}{4}$ by 16 and 12x separately.

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

A: To isolate the variable in a linear equation, you need to get rid of the constant term on the left-hand side of the equation. This can be done by subtracting the constant term from 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 get rid of the coefficient of the variable. This can be done by dividing both sides of the equation by the coefficient.

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

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

Not distributing the fraction correctly
Not isolating the variable correctly
Not solving for the variable correctly

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by trying the following problems:

$\frac{1}{2}(24 + 16x) = 36$
$\frac{3}{4}(12 + 20x) = 45$
$\frac{2}{3}(18 + 12x) = 24$

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

A: Linear equations have many real-world applications, including:

Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: How can I apply linear equations to real-world problems?

A: To apply linear equations to real-world problems, you need to identify the variables and constants in the problem, and then use the distributive property and other mathematical properties to solve for the variable.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, readers should be able to solve similar linear equations with ease. Remember to distribute the fraction correctly, isolate the variable correctly, and solve for the variable correctly. With practice and patience, readers should be able to apply linear equations to real-world problems and become proficient in solving them.

Additional Resources

For additional resources on solving linear equations, including practice problems and real-world applications, please visit the following websites:

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

Final Tips

Practice solving linear equations regularly to build your skills and confidence.
Use online resources, such as Khan Academy and Mathway, to get additional help and practice problems.
Apply linear equations to real-world problems to see the practical applications of the concept.