Examine The System Of Equations:$\[ \begin{array}{l} 4.2x + 8y = 41.8 \\ -4.2x + Y = 19.4 \end{array} \\]Use The Linear Combination Method To Solve The System Of Equations. What Is The Value Of $x$?A. -3 B. -1 C. 1.7 D. 6.8

Introduction

In this article, we will examine a system of two linear equations and use the linear combination method to solve for the value of $x$. The system of equations is given as:

{ \begin{array}{l} 4.2x + 8y = 41.8 \\ -4.2x + y = 19.4 \end{array} \}

Understanding the Linear Combination Method

The linear combination method is a technique used to solve a system of linear equations by combining the equations in a way that eliminates one of the variables. This method involves multiplying one or both of the equations by a constant and then adding or subtracting the resulting equations to eliminate one of the variables.

Step 1: Multiply the Second Equation by 8

To eliminate the variable $y$, we need to multiply the second equation by 8 so that the coefficients of $y$ in both equations are the same. This gives us:

{ \begin{array}{l} 4.2x + 8y = 41.8 \\ -33.6x + 8y = 155.2 \end{array} \}

Step 2: Subtract the First Equation from the Second Equation

Now, we subtract the first equation from the second equation to eliminate the variable $y$. This gives us:

{ \begin{array}{l} 4.2x + 8y = 41.8 \\ -33.6x + 8y = 155.2 \end{array} \}

$-33.6x - 4.2x = 155.2 - 41.8$

$-37.8x = 113.4$

Step 3: Solve for $x$

Now, we can solve for $x$ by dividing both sides of the equation by -37.8. This gives us:

$x = \frac{113.4}{-37.8}$

$x = -3$

Conclusion

In this article, we used the linear combination method to solve a system of two linear equations. We multiplied the second equation by 8 to eliminate the variable $y$, and then subtracted the first equation from the second equation to solve for $x$. The value of $x$ is -3.

Discussion

The linear combination method is a powerful technique for solving systems of linear equations. It involves combining the equations in a way that eliminates one of the variables, making it easier to solve for the other variable. In this case, we used the linear combination method to solve for $x$, and the value of $x$ is -3.

Final Answer

Introduction

In our previous article, we used the linear combination method to solve a system of two linear equations. In this article, we will answer some frequently asked questions about the linear combination method and provide additional examples to help you understand the concept better.

Q: What is the linear combination method?

A: The linear combination method is a technique used to solve a system of linear equations by combining the equations in a way that eliminates one of the variables. This method involves multiplying one or both of the equations by a constant and then adding or subtracting the resulting equations to eliminate one of the variables.

Q: How do I know which equation to multiply by a constant?

A: To determine which equation to multiply by a constant, you need to look at the coefficients of the variables in both equations. You want to multiply the equation by a constant that will make the coefficients of the variables the same. For example, if the coefficients of $y$ in both equations are different, you can multiply the equation with the smaller coefficient by a constant that will make the coefficients the same.

Q: What if I have a system of three or more linear equations?

A: If you have a system of three or more linear equations, you can use the linear combination method to solve for one or two variables, and then use substitution or elimination to solve for the remaining variables.

Q: Can I use the linear combination method to solve a system of nonlinear equations?

A: No, the linear combination method is only used to solve systems of linear equations. If you have a system of nonlinear equations, you will need to use a different method, such as substitution or elimination, to solve for the variables.

Q: What are some common mistakes to avoid when using the linear combination method?

A: Some common mistakes to avoid when using the linear combination method include:

• Multiplying the wrong equation by a constant
• Not checking if the coefficients of the variables are the same after multiplying
• Not subtracting or adding the equations correctly
• Not solving for the correct variable

Q: Can I use the linear combination method to solve a system of equations with fractions?

A: Yes, you can use the linear combination method to solve a system of equations with fractions. However, you will need to multiply the fractions by a common denominator to eliminate the fractions.

Q: What are some real-world applications of the linear combination method?

A: The linear combination method has many real-world applications, including:

• Solving systems of equations in physics and engineering
• Finding the intersection of two or more lines in geometry
• Solving systems of equations in economics and finance
• Finding the solution to a system of equations in computer science

Conclusion

In this article, we answered some frequently asked questions about the linear combination method and provided additional examples to help you understand the concept better. We also discussed some common mistakes to avoid and real-world applications of the linear combination method.

Final Answer

The final answer is $\boxed{yes}$, the linear combination method is a powerful technique for solving systems of linear equations, and it has many real-world applications.