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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. The equation we will be solving is $\frac{1}{4}(16+12x)=28$. We will break down the solution into manageable steps, making it easy for readers to follow along.

Step 1: Distribute the $\frac{1}{4}$

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

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

Using the distributive property, we can rewrite the equation as:

$4 + 3x = 28$

Step 2: Isolate 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. We can do this by subtracting 4 from both sides of the equation.

$3x = 28 - 4$

$3x = 24$

Step 3: Solve for $x$

Now that we have isolated the variable, we can solve for $x$. To do this, we need to divide both sides of the equation by 3.

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

$x = 8$

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 easy for readers to follow along. By distributing the $\frac{1}{4}$, isolating the variable, and solving for $x$, we were able to find the solution to the equation.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS).
• Use the distributive property to simplify expressions and make it easier to solve for the variable.
• Isolate the variable by getting rid of the constant term on the left-hand side of the equation.
• Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Real-World Applications

Linear equations have numerous real-world applications. For example, they can be used to model population growth, financial transactions, and scientific experiments. By understanding how to solve linear equations, individuals can make informed decisions and solve problems in various fields.

Common Mistakes to Avoid

• Failing to distribute the coefficient to the terms inside the parentheses.
• Not isolating the variable by getting rid of the constant term on the left-hand side of the equation.
• Not solving for the variable by dividing both sides of the equation by the coefficient of the variable.

Conclusion

Introduction

In our previous article, we solved the linear equation $\frac{1}{4}(16+12x)=28$ using the distributive property. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to distribute a coefficient to the terms inside the parentheses. It states that for any numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

Q: How do I apply the distributive property to solve a linear equation?

A: To apply the distributive property, you need to multiply the coefficient to each term inside the parentheses. For example, if you have the equation $\frac{1}{4}(16+12x)$, you would multiply the $\frac{1}{4}$ to each term inside the parentheses:

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

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when solving an equation. The order of operations is:

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

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

A: To isolate the variable, you need to get rid of the constant term on the left-hand side of the equation. You can do this by subtracting the constant term from both sides of the equation. For example, if you have the equation $3x + 4 = 12$, you would subtract 4 from both sides of the equation:

$3x + 4 - 4 = 12 - 4$

$3x = 8$

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. For example, the equation $2x + 3 = 5$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

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

A: To solve a linear equation with fractions, you need to follow the same steps as solving a linear equation with whole numbers. However, you may need to multiply both sides of the equation by the denominator of the fraction to eliminate the fraction. For example, if you have the equation $\frac{1}{2}x + 3 = 5$, you would multiply both sides of the equation by 2 to eliminate the fraction:

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

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

$x + 6 = 10$

$x = 4$

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, individuals can solve linear equations with ease. Remember to distribute the coefficient, isolate the variable, and solve for the variable to find the solution to the equation. With practice and patience, individuals can become proficient in solving linear equations and apply their skills to real-world problems.