Solve The Following System Of Equations By Elimination, If A Solution Exists:${ \begin{cases} 5x + 15y = 25 \ 6y - 10 = -2x \end{cases} }$Select The Correct Choice Below And Fill In Any Answer Boxes In Your Choice.A. The Solution To The

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. One of the methods used to solve a system of equations is the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables and then solving for the other variable. In this article, we will solve a system of equations by elimination and discuss the steps involved in this process.

The System of Equations

The given system of equations is:

${ \begin{cases} 5x + 15y = 25 \\ 6y - 10 = -2x \end{cases} }$

To solve this system of equations by elimination, we need to follow the steps outlined below.

Step 1: Multiply the Equations by Necessary Multiples

The first step in solving a system of equations by elimination is to multiply the equations by necessary multiples such that the coefficients of one of the variables (either x or y) are the same in both equations. In this case, we can multiply the first equation by 2 and the second equation by 5 to make the coefficients of x the same.

${ 2(5x + 15y) = 2(25) }$ ${ 10x + 30y = 50 }$

${ 5(6y - 10) = 5(-2x) }$ ${ 30y - 50 = -10x }$

Step 2: Add or Subtract the Equations

Once we have multiplied the equations by necessary multiples, we can add or subtract the equations to eliminate one of the variables. In this case, we can add the two equations to eliminate the variable x.

${ (10x + 30y) + (-10x + 30y) = 50 + (-50) }$ ${ 60y = 0 }$

Step 3: Solve for the Variable

Now that we have eliminated the variable x, we can solve for the variable y. In this case, we can divide both sides of the equation by 60 to solve for y.

${ y = \frac{0}{60} }$ ${ y = 0 }$

Step 4: Substitute the Value of the Variable into One of the Original Equations

Now that we have found the value of y, we can substitute this value into one of the original equations to solve for the variable x. In this case, we can substitute y = 0 into the first equation.

${ 5x + 15(0) = 25 }$ ${ 5x = 25 }$ ${ x = \frac{25}{5} }$ ${ x = 5 }$

Conclusion

In this article, we solved a system of equations by elimination. We multiplied the equations by necessary multiples, added or subtracted the equations to eliminate one of the variables, solved for the variable, and then substituted the value of the variable into one of the original equations to solve for the other variable. The solution to the system of equations is x = 5 and y = 0.

Discussion

The elimination method is a powerful tool for solving systems of equations. By multiplying the equations by necessary multiples and adding or subtracting the equations, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when the coefficients of one of the variables are the same in both equations.

Example Problems

1. Solve the system of equations by elimination:

${ \begin{cases} 2x + 3y = 7 \\ 4x - 2y = -3 \end{cases} }$

2. Solve the system of equations by elimination:

${ \begin{cases} x + 2y = 4 \\ 3x - 4y = -5 \end{cases} }$

Solutions

1. The solution to the system of equations is x = 1 and y = 1.

2. The solution to the system of equations is x = 3 and y = -1.

Conclusion

Introduction

In our previous article, we solved a system of equations by elimination and discussed the steps involved in this process. In this article, we will answer some frequently asked questions about solving a system of equations by elimination.

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: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in both equations. If the coefficients of one of the variables are the same in both equations, you can eliminate that variable.

Q: What if the coefficients of the variables are not the same in both equations?

A: If the coefficients of the variables are not the same in both equations, you need to multiply the equations by necessary multiples to make the coefficients of one of the variables the same.

Q: How do I add or subtract the equations to eliminate one of the variables?

A: To add or subtract the equations, you need to multiply the equations by necessary multiples to make the coefficients of one of the variables the same, and then add or subtract the equations.

Q: What if I get a false statement when I add or subtract the equations?

A: If you get a false statement when you add or subtract the equations, it means that the system of equations has no solution.

Q: How do I know if the system of equations has a solution?

A: To determine if the system of equations has a solution, you need to check if the resulting equation after adding or subtracting the equations is a true statement.

Q: What if I get a solution, but it is not a whole number?

A: If you get a solution, but it is not a whole number, it means that the solution is a decimal or a fraction.

Q: Can I use the elimination method to solve a system of three or more equations?

A: Yes, you can use the elimination method to solve a system of three or more equations. However, it may be more complicated and require more steps.

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

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

• Not multiplying the equations by necessary multiples to make the coefficients of one of the variables the same.
• Not adding or subtracting the equations correctly.
• Not checking if the resulting equation after adding or subtracting the equations is a true statement.

Conclusion

In this article, we answered some frequently asked questions about solving a system of equations by elimination. We hope that this article has helped to clarify any confusion and provide a better understanding of the elimination method.

Example Problems

1. Solve the system of equations by elimination:

${ \begin{cases} 2x + 3y = 7 \\ 4x - 2y = -3 \end{cases} }$

2. Solve the system of equations by elimination:

${ \begin{cases} x + 2y = 4 \\ 3x - 4y = -5 \end{cases} }$

Solutions

1. The solution to the system of equations is x = 1 and y = 1.

2. The solution to the system of equations is x = 3 and y = -1.

Tips and Tricks

• Make sure to multiply the equations by necessary multiples to make the coefficients of one of the variables the same.
• Check if the resulting equation after adding or subtracting the equations is a true statement.
• Use the elimination method to solve a system of three or more equations, but be aware that it may be more complicated and require more steps.

Conclusion

In this article, we provided a Q&A section on solving a system of equations by elimination. We hope that this article has helped to clarify any confusion and provide a better understanding of the elimination method.