How Can 1 4 X − 3 = 1 2 X + 8 \frac{1}{4}x - 3 = \frac{1}{2}x + 8 4 1 ​ X − 3 = 2 1 ​ X + 8 Be Set Up As A System Of Equations?A. 4 Y + 4 X = − 12 4y + 4x = -12 4 Y + 4 X = − 12 B. 2 Y + 2 X = 16 2y + 2x = 16 2 Y + 2 X = 16 C. 4 Y − X = − 12 4y - X = -12 4 Y − X = − 12 D. 2 Y − X = 16 2y - X = 16 2 Y − X = 16 E. $[ \begin{align*} 4y + X &= -12 \ 2y + X

Introduction

In mathematics, solving linear equations is a fundamental concept that forms the basis of various mathematical operations. One of the ways to solve linear equations is by setting up a system of equations. A system of equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will explore how to set up a system of equations using the given linear equation $\frac{1}{4}x - 3 = \frac{1}{2}x + 8$.

Understanding the Given Equation

The given equation is $\frac{1}{4}x - 3 = \frac{1}{2}x + 8$. To set up a system of equations, we need to isolate the variables in each equation. Let's start by simplifying the equation.

Simplifying the Equation

To simplify the equation, we can multiply both sides by the least common multiple (LCM) of the denominators, which is 4.

\frac{1}{4}x - 3 = \frac{1}{2}x + 8
\implies 4 \times \left(\frac{1}{4}x - 3\right) = 4 \times \left(\frac{1}{2}x + 8\right)
\implies x - 12 = 2x + 32

Isolating the Variables

Now that we have simplified the equation, let's isolate the variables.

x - 12 = 2x + 32
\implies -12 = x + 32
\implies -44 = x

However, we are not interested in finding the value of x. Instead, we want to set up a system of equations. To do this, we need to rewrite the equation in the form of a system of equations.

Setting Up a System of Equations

Let's rewrite the equation in the form of a system of equations.

\frac{1}{4}x - 3 = \frac{1}{2}x + 8
\implies 4 \times \left(\frac{1}{4}x - 3\right) = 4 \times \left(\frac{1}{2}x + 8\right)
\implies x - 12 = 2x + 32
\implies -12 = x + 32
\implies -44 = x

However, we can rewrite the equation in the form of a system of equations by multiplying both sides by 4 and then subtracting 2x from both sides.

4 \times \left(\frac{1}{4}x - 3\right) = 4 \times \left(\frac{1}{2}x + 8\right)
\implies x - 12 = 2x + 32
\implies -12 = x + 32
\implies -44 = x
\implies 4y + x = -12
\implies 2y + x = 16

Solving the System of Equations

Now that we have set up the system of equations, we can solve it using substitution or elimination. Let's use the elimination method.

4y + x = -12
\implies 2y + x = 16
\implies 2y + x - (2y + x) = 16 - (-12)
\implies 0 = 28

However, this is not a valid solution. The problem is that we have not isolated the variables correctly. Let's try again.

4y + x = -12
\implies 2y + x = 16
\implies 4y + x - (2y + x) = -12 - 16
\implies 2y = -28
\implies y = -14

Now that we have found the value of y, we can substitute it into one of the equations to find the value of x.

4y + x = -12
\implies 4(-14) + x = -12
\implies -56 + x = -12
\implies x = 44

Conclusion

In this article, we have explored how to set up a system of equations using the given linear equation $\frac{1}{4}x - 3 = \frac{1}{2}x + 8$. We have shown that the correct system of equations is:

4y + x = -12
\implies 2y + x = 16

We have also solved the system of equations using the elimination method and found the values of x and y.

Final Answer

The final answer is:

• $4y + x = -12$
• $2y + x = 16$

Q: What is a system of equations?

A: A system of equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: How do I set up a system of equations?

A: To set up a system of equations, you need to isolate the variables in each equation. You can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, or by using other algebraic techniques such as addition, subtraction, multiplication, and division.

Q: What are the steps to solve a system of equations?

A: The steps to solve a system of equations are:

1. Isolate the variables in each equation.
2. Multiply both sides of the equation by the LCM of the denominators, or use other algebraic techniques to simplify the equation.
3. Use the elimination method or substitution method to solve the system of equations.
4. Check the solution by plugging it back into the original equations.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of equations by adding or subtracting the equations to eliminate one of the variables.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting one of the variables from one equation into the other equation.

Q: How do I choose between the elimination method and the substitution method?

A: You can choose between the elimination method and the substitution method based on the form of the equations. If the coefficients of one of the variables are the same in both equations, it is easier to use the elimination method. If the coefficients of one of the variables are different in both equations, it is easier to use the substitution method.

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

A: Some common mistakes to avoid when solving a system of equations include:

• Not isolating the variables correctly
• Not multiplying both sides of the equation by the LCM of the denominators
• Not using the correct algebraic techniques to simplify the equation
• Not checking the solution by plugging it back into the original equations

Q: How do I check my solution?

A: To check your solution, you need to plug it back into the original equations and make sure that it satisfies both equations.

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

A: Systems of equations have many real-world applications, including:

• Physics: Systems of equations are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Systems of equations are used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
• Economics: Systems of equations are used to model economic systems and make predictions about the behavior of economic variables.
• Computer Science: Systems of equations are used to solve problems in computer science such as graph theory, network flow, and optimization.

Conclusion

In this article, we have explored the concept of systems of equations and how to solve them using the elimination method and the substitution method. We have also discussed some common mistakes to avoid when solving a system of equations and how to check your solution. Systems of equations have many real-world applications and are an important tool in many fields.