Solve The Equation $\frac{1}{4}(4x - 24) + X = 14$.1. Distribute The $\frac{1}{4}$ To The Quantity: $\[ X - 6 + X = 14 \\]2. Combine The Like Terms: $\[ 2x - 6 = 14 \\]3. Add 6 To Both Sides: $\[ 2x = 20

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, $\frac{1}{4}(4x - 24) + x = 14$, using a step-by-step approach. We will break down the solution into manageable parts, making it easier to understand and follow.

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

The first step in solving the equation is to distribute the $\frac{1}{4}$ to the quantity inside the parentheses. This means multiplying the $\frac{1}{4}$ by each term inside the parentheses.

$\frac{1}{4}(4x - 24) + x = 14$

To distribute the $\frac{1}{4}$, we multiply it by each term inside the parentheses:

$\frac{1}{4} \cdot 4x = x$

$\frac{1}{4} \cdot (-24) = -6$

So, the equation becomes:

$x - 6 + x = 14$

Step 2: Combine the Like Terms

Now that we have distributed the $\frac{1}{4}$, we can combine the like terms. In this case, the like terms are the two $x$ terms.

$x - 6 + x = 14$

Combining the like terms, we get:

$2x - 6 = 14$

Step 3: Add 6 to Both Sides

The next step is to isolate the variable $x$ by adding 6 to both sides of the equation.

$2x - 6 = 14$

Adding 6 to both sides, we get:

$2x = 20$

Step 4: Divide Both Sides by 2

Finally, we need to solve for $x$ by dividing both sides of the equation by 2.

$2x = 20$

Dividing both sides by 2, we get:

$x = 10$

Conclusion

In this article, we have solved the linear equation $\frac{1}{4}(4x - 24) + x = 14$ using a step-by-step approach. We distributed the $\frac{1}{4}$ to the quantity inside the parentheses, combined the like terms, added 6 to both sides, and finally divided both sides by 2 to solve for $x$. By following these steps, we have successfully solved the equation and found the value of $x$.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When distributing a coefficient to a quantity, make sure to multiply each term inside the parentheses by the coefficient.
• When combining like terms, make sure to combine the coefficients of the like terms.
• When adding or subtracting a constant to both sides of an equation, make sure to add or subtract the constant to both sides.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
• Economics: Linear equations are used to model economic systems, including supply and demand curves, and to make predictions about economic trends.

Common Mistakes

• When solving linear equations, it's easy to make mistakes, especially when distributing coefficients or combining like terms. Make sure to double-check your work and use a calculator to check your answers.
• When adding or subtracting a constant to both sides of an equation, make sure to add or subtract the constant to both sides.
• When dividing both sides of an equation by a coefficient, make sure to divide both sides by the coefficient.

Conclusion

Introduction

In our previous article, we covered the step-by-step process of solving a linear equation, $\frac{1}{4}(4x - 24) + x = 14$. We distributed the $\frac{1}{4}$ to the quantity inside the parentheses, combined the like terms, added 6 to both sides, and finally divided both sides by 2 to solve for $x$. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation in which the variable is not raised to a power greater than 1.

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

A: To determine if an equation is linear, look for the highest power of the variable. If the highest power is 1, then the equation is linear.

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 distribute a coefficient to a quantity?

A: To distribute a coefficient to a quantity, multiply the coefficient by each term inside the parentheses.

Q: What is the difference between a coefficient and a constant?

A: A coefficient is a number that is multiplied by a variable, while a constant is a number that is not multiplied by a variable.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms.

Q: What is the difference between adding and subtracting a constant to both sides of an equation?

A: When adding a constant to both sides of an equation, you are adding the same value to both sides. When subtracting a constant from both sides of an equation, you are subtracting the same value from both sides.

Q: How do I divide both sides of an equation by a coefficient?

A: To divide both sides of an equation by a coefficient, divide both sides 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 coefficients to quantities
• Not combining like terms
• Not adding or subtracting constants to both sides of an equation
• Not dividing both sides of an equation by a coefficient

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

• Using online resources, such as Khan Academy or Mathway
• Working with a tutor or teacher
• Practicing with worksheets or exercises
• Solving real-world problems that involve linear equations

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations with ease. Remember to distribute coefficients, combine like terms, add or subtract constants, and divide both sides by coefficients to solve for the variable. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.