What Is The Value Of $x$ In The Equation Below? 1 3 ( 12 X − 24 ) = 16 \frac{1}{3}(12x - 24) = 16 3 1 ​ ( 12 X − 24 ) = 16 A. 2 B. 6 C. 8 D. 10

Understanding the Equation

The given equation is $\frac{1}{3}(12x - 24) = 16$. To find the value of $x$, we need to isolate the variable $x$ on one side of the equation. This involves simplifying the equation and performing algebraic operations to solve for $x$.

Simplifying the Equation

The first step is to simplify the left-hand side of the equation by distributing the $\frac{1}{3}$ to the terms inside the parentheses. This gives us:

$\frac{1}{3}(12x - 24) = \frac{12x}{3} - \frac{24}{3}$

Combining Like Terms

Next, we can combine the like terms on the left-hand side of the equation:

$\frac{12x}{3} - \frac{24}{3} = 4x - 8$

Setting Up the Equation

Now that we have simplified the left-hand side of the equation, we can set it equal to the right-hand side:

$4x - 8 = 16$

Solving for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by adding $8$ to both sides of the equation:

$4x - 8 + 8 = 16 + 8$

This simplifies to:

$4x = 24$

Dividing Both Sides

Next, we can divide both sides of the equation by $4$ to solve for $x$:

$\frac{4x}{4} = \frac{24}{4}$

This simplifies to:

$x = 6$

Conclusion

Therefore, the value of $x$ in the equation $\frac{1}{3}(12x - 24) = 16$ is $6$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Simplify the left-hand side of the equation by distributing the $\frac{1}{3}$ to the terms inside the parentheses.
2. Combine the like terms on the left-hand side of the equation.
3. Set up the equation by setting the simplified left-hand side equal to the right-hand side.
4. Add $8$ to both sides of the equation to isolate the variable $x$.
5. Divide both sides of the equation by $4$ to solve for $x$.

Final Answer

The final answer is $\boxed{6}$.

Why is this the Correct Answer?

This is the correct answer because we have isolated the variable $x$ on one side of the equation and solved for its value. The equation is now in the form $x = 6$, which means that the value of $x$ is indeed $6$.

What is the Importance of Solving Equations?

Solving equations is an important skill in mathematics because it allows us to find the value of unknown variables. In this case, we were able to find the value of $x$ by simplifying the equation and performing algebraic operations. This skill is essential in many areas of mathematics, including algebra, geometry, and calculus.

How to Solve Equations

To solve equations, we need to follow a series of steps. These steps include:

1. Simplifying the equation by combining like terms and distributing coefficients.
2. Isolating the variable by adding or subtracting the same value to both sides of the equation.
3. Dividing both sides of the equation by a coefficient to solve for the variable.

Tips for Solving Equations

Here are some tips for solving equations:

1. Read the equation carefully and identify the variable and the constant.
2. Simplify the equation by combining like terms and distributing coefficients.
3. Isolate the variable by adding or subtracting the same value to both sides of the equation.
4. Divide both sides of the equation by a coefficient to solve for the variable.
5. Check your answer by plugging it back into the original equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving equations:

1. Not simplifying the equation by combining like terms and distributing coefficients.
2. Not isolating the variable by adding or subtracting the same value to both sides of the equation.
3. Not dividing both sides of the equation by a coefficient to solve for the variable.
4. Not checking your answer by plugging it back into the original equation.

Conclusion

In conclusion, solving equations is an important skill in mathematics that allows us to find the value of unknown variables. By following a series of steps and avoiding common mistakes, we can solve equations and find the value of variables. In this case, we were able to find the value of $x$ in the equation $\frac{1}{3}(12x - 24) = 16$ by simplifying the equation and performing algebraic operations. The final answer is $\boxed{6}$.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), which are separated by an equal sign (=).

Q: What is the purpose of solving equations?

A: The purpose of solving equations is to find the value of the unknown variable(s) in the equation. This is done by isolating the variable(s) on one side of the equation and solving for its value.

Q: What are the steps to solve an equation?

A: The steps to solve an equation are:

1. Simplify the equation by combining like terms and distributing coefficients.
2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
3. Divide both sides of the equation by a coefficient to solve for the variable.
4. Check your answer by plugging it back into the original equation.

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(s) is 1. For example, 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(s) is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

1. Substitution method: Substitute the expression for one variable from one equation into the other equation.
2. Elimination method: Add or subtract the equations to eliminate one variable.
3. Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the importance of solving equations in real-life situations?

A: Solving equations is an important skill in many real-life situations, such as:

1. Science: Solving equations is used to model real-world phenomena, such as the motion of objects and the behavior of populations.
2. Engineering: Solving equations is used to design and optimize systems, such as bridges and buildings.
3. Economics: Solving equations is used to model economic systems and make predictions about future trends.

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

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

1. Not simplifying the equation by combining like terms and distributing coefficients.
2. Not isolating the variable by adding or subtracting the same value to both sides of the equation.
3. Not dividing both sides of the equation by a coefficient to solve for the variable.
4. Not checking your answer by plugging it back into the original equation.

Q: How can I practice solving equations?

A: You can practice solving equations by:

1. Working on math problems and exercises.
2. Using online resources, such as math websites and apps.
3. Joining a study group or finding a study partner.
4. Taking online courses or watching video tutorials.

Q: What are some resources for learning more about solving equations?

A: Some resources for learning more about solving equations include:

1. Math textbooks and workbooks.
2. Online math resources, such as Khan Academy and Mathway.
3. Math apps and software, such as Wolfram Alpha and Mathematica.
4. Online courses and tutorials, such as Coursera and edX.

Q: How can I apply solving equations to real-life situations?

A: You can apply solving equations to real-life situations by:

1. Using math to model real-world phenomena, such as the motion of objects and the behavior of populations.
2. Designing and optimizing systems, such as bridges and buildings.
3. Modeling economic systems and making predictions about future trends.
4. Solving problems in science, engineering, and other fields that require mathematical modeling.