How Can You Eliminate The Y Y Y -terms In This System? \begin{align*} 4x - 8y &= 20 \\ 3x + 4y &= 5 \end{align*} Multiply The Equations By:A. 2B. 3C. 4D. 5

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Introduction

In this article, we will explore the process of eliminating $y$-terms in a system of linear equations. This is a crucial step in solving systems of linear equations, as it allows us to reduce the number of variables and make the system easier to solve. We will use the given system of linear equations as an example to demonstrate the process.

The Given System of Linear Equations

The given system of linear equations is:

$$\begin{align*} 4x - 8y &= 20 \\ 3x + 4y &= 5 \end{align*}$$

Multiplying the Equations by a Constant

To eliminate the $y$-terms, we need to multiply the equations by a constant such that the coefficients of $y$ in both equations are the same. Let's consider the options:

• Option A: Multiply the equations by 2
• Option B: Multiply the equations by 3
• Option C: Multiply the equations by 4
• Option D: Multiply the equations by 5

We will analyze each option and determine which one is the best choice.

Option A: Multiply the Equations by 2

If we multiply the first equation by 2, we get:

$$\begin{align*} 8x - 16y &= 40 \\ 3x + 4y &= 5 \end{align*}$$

If we multiply the second equation by 2, we get:

$$\begin{align*} 6x + 8y &= 10 \\ 3x + 4y &= 5 \end{align*}$$

In this case, the coefficients of $y$ in both equations are not the same, so this option is not the best choice.

Option B: Multiply the Equations by 3

If we multiply the first equation by 3, we get:

$$\begin{align*} 12x - 24y &= 60 \\ 3x + 4y &= 5 \end{align*}$$

If we multiply the second equation by 3, we get:

$$\begin{align*} 9x + 12y &= 15 \\ 3x + 4y &= 5 \end{align*}$$

In this case, the coefficients of $y$ in both equations are not the same, so this option is not the best choice.

Option C: Multiply the Equations by 4

If we multiply the first equation by 4, we get:

$$\begin{align*} 16x - 32y &= 80 \\ 3x + 4y &= 5 \end{align*}$$

If we multiply the second equation by 4, we get:

$$\begin{align*} 12x + 16y &= 20 \\ 3x + 4y &= 5 \end{align*}$$

In this case, the coefficients of $y$ in both equations are not the same, so this option is not the best choice.

Option D: Multiply the Equations by 5

If we multiply the first equation by 5, we get:

$$\begin{align*} 20x - 40y &= 100 \\ 3x + 4y &= 5 \end{align*}$$

If we multiply the second equation by 5, we get:

$$\begin{align*} 15x + 20y &= 25 \\ 3x + 4y &= 5 \end{align*}$$

In this case, the coefficients of $y$ in both equations are the same, so this option is the best choice.

Eliminating the $y$-terms

Now that we have multiplied the equations by 5, we can eliminate the $y$-terms by adding the two equations together:

$$\begin{align*} 20x - 40y &= 100 \\ 3x + 4y &= 5 \end{align*}$$

Adding the two equations together, we get:

$$\begin{align*} 23x - 36y &= 105 \end{align*}$$

However, we want to eliminate the $y$-terms, so we need to get rid of the $y$-term in the equation. We can do this by multiplying the second equation by 9 and adding it to the first equation:

$$\begin{align*} 20x - 40y &= 100 \\ 27x + 36y &= 45 \end{align*}$$

Adding the two equations together, we get:

$$\begin{align*} 47x &= 145 \end{align*}$$

Now we can solve for $x$ by dividing both sides of the equation by 47:

$$\begin{align*} x &= \frac{145}{47} \\ x &= 3.0851 \end{align*}$$

Conclusion

In this article, we have demonstrated how to eliminate the $y$-terms in a system of linear equations. We used the given system of linear equations as an example and multiplied the equations by a constant to make the coefficients of $y$ in both equations the same. We then added the two equations together to eliminate the $y$-terms and solved for $x$. This process is a crucial step in solving systems of linear equations and can be applied to a wide range of problems.

Final Answer

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Introduction

In our previous article, we demonstrated how to eliminate the $y$-terms in a system of linear equations. In this article, we will answer some frequently asked questions about this process.

Q: What is the purpose of eliminating $y$-terms in a system of linear equations?

A: The purpose of eliminating $y$-terms in a system of linear equations is to reduce the number of variables and make the system easier to solve. By eliminating the $y$-terms, we can solve for one variable and then substitute that value into the other equation to solve for the other variable.

Q: How do I know which constant to multiply the equations by?

A: To determine which constant to multiply the equations by, you need to find a constant that will make the coefficients of $y$ in both equations the same. This can be done by trial and error or by using a calculator to find the least common multiple (LCM) of the coefficients of $y$.

Q: What if the coefficients of $y$ in both equations are not the same after multiplying by a constant?

A: If the coefficients of $y$ in both equations are not the same after multiplying by a constant, you need to try a different constant. You can try multiplying the equations by a different constant or using a different method to eliminate the $y$-terms.

Q: Can I eliminate $y$-terms in a system of linear equations with more than two variables?

A: Yes, you can eliminate $y$-terms in a system of linear equations with more than two variables. However, the process is more complex and may require the use of matrices or other advanced techniques.

Q: What if I make a mistake when eliminating $y$-terms?

A: If you make a mistake when eliminating $y$-terms, you may end up with an incorrect solution. To avoid this, it's essential to double-check your work and make sure that you have eliminated the $y$-terms correctly.

Q: Are there any other methods for eliminating $y$-terms in a system of linear equations?

A: Yes, there are other methods for eliminating $y$-terms in a system of linear equations. Some of these methods include:

• Using matrices to solve the system of linear equations
• Using the substitution method to solve the system of linear equations
• Using the elimination method to solve the system of linear equations

Conclusion

In this article, we have answered some frequently asked questions about eliminating $y$-terms in a system of linear equations. We have also provided some tips and tricks for making the process easier and avoiding common mistakes.

Final Answer

The final answer is $\boxed{yes}$, there are many methods for eliminating $y$-terms in a system of linear equations.

Additional Resources

For more information on eliminating $y$-terms in a system of linear equations, please see the following resources:

• Mathway: A online math problem solver that can help you solve systems of linear equations.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on systems of linear equations.
• Wolfram Alpha: A online calculator that can help you solve systems of linear equations and other math problems.

FAQs

Q: What is the difference between the elimination method and the substitution method?

A: The elimination method and the substitution method are two different methods for solving systems of linear equations. The elimination method involves eliminating one variable by adding or subtracting the equations, while the substitution method involves substituting one variable into the other equation to solve for the other variable.

Q: Can I use the elimination method to solve a system of linear equations with more than two variables?

A: Yes, you can use the elimination method to solve a system of linear equations with more than two variables. However, the process is more complex and may require the use of matrices or other advanced techniques.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

Q: Can I use a calculator to find the LCM of two numbers?

A: Yes, you can use a calculator to find the LCM of two numbers. Most calculators have a built-in function for finding the LCM.

Conclusion

In this article, we have answered some frequently asked questions about eliminating $y$-terms in a system of linear equations. We have also provided some tips and tricks for making the process easier and avoiding common mistakes.