Given The System Of Equations:${ \begin{cases} 2x + Y = 3 \ 3x - 4y = 10 \end{cases} }$Verify The Solution { X = 2 $}$ And { Y = -1 $}$.

Introduction

In mathematics, a system of equations is a set of equations that are all true at the same time. Solving a system of equations involves finding the values of the variables that make all the equations true. In this article, we will verify the solution to a system of two linear equations in two variables. We will use the given solution to check if it satisfies both equations.

The System of Equations

The system of equations is given as:

${ \begin{cases} 2x + y = 3 \\ 3x - 4y = 10 \end{cases} }$

The Given Solution

The given solution is:

${ x = 2 }$

${ y = -1 }$

Verifying the Solution

To verify the solution, we need to substitute the values of x and y into both equations and check if they are true.

Equation 1: 2x + y = 3

Substituting x = 2 and y = -1 into the first equation, we get:

${ 2(2) + (-1) = 3 }$

${ 4 - 1 = 3 }$

${ 3 = 3 }$

This equation is true.

Equation 2: 3x - 4y = 10

Substituting x = 2 and y = -1 into the second equation, we get:

${ 3(2) - 4(-1) = 10 }$

${ 6 + 4 = 10 }$

${ 10 = 10 }$

This equation is also true.

Conclusion

Since both equations are true when x = 2 and y = -1, we can conclude that the given solution is correct.

Discussion

In this article, we verified the solution to a system of two linear equations in two variables. We used the given solution to check if it satisfies both equations. This is an important step in solving systems of equations, as it ensures that the solution is correct.

Tips and Tricks

When verifying a solution to a system of equations, make sure to substitute the values of the variables into all the equations. This will help you catch any mistakes and ensure that the solution is correct.

Example Problems

Here are a few example problems to try:

1. Verify the solution x = 3 and y = 2 to the system of equations:

${ \begin{cases} x + 2y = 5 \\ 2x - 3y = 1 \end{cases} }$

2. Verify the solution x = -2 and y = 3 to the system of equations:

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

Solutions

1. To verify the solution x = 3 and y = 2, substitute the values into both equations:

Equation 1: x + 2y = 5

Substituting x = 3 and y = 2 into the first equation, we get:

${ 3 + 2(2) = 5 }$

${ 3 + 4 = 5 }$

${ 7 = 5 }$

This equation is not true.

Equation 2: 2x - 3y = 1

Substituting x = 3 and y = 2 into the second equation, we get:

${ 2(3) - 3(2) = 1 }$

${ 6 - 6 = 1 }$

${ 0 = 1 }$

This equation is not true.

Since both equations are not true when x = 3 and y = 2, we can conclude that the given solution is incorrect.

2. To verify the solution x = -2 and y = 3, substitute the values into both equations:

Equation 1: 2x + 3y = 7

Substituting x = -2 and y = 3 into the first equation, we get:

${ 2(-2) + 3(3) = 7 }$

${ -4 + 9 = 7 }$

${ 5 = 7 }$

This equation is not true.

Equation 2: x - 2y = -3

Substituting x = -2 and y = 3 into the second equation, we get:

${ (-2) - 2(3) = -3 }$

${ -2 - 6 = -3 }$

${ -8 = -3 }$

This equation is not true.

Since both equations are not true when x = -2 and y = 3, we can conclude that the given solution is incorrect.

Conclusion

Introduction

In our previous article, we discussed how to verify the solution to a system of two linear equations in two variables. We used the given solution to check if it satisfies both equations. In this article, we will provide a Q&A guide to help you understand the concept of verifying solutions to systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of equations that are all true at the same time. Solving a system of equations involves finding the values of the variables that make all the equations true.

Q: How do I verify a solution to a system of equations?

A: To verify a solution to a system of equations, you need to substitute the values of the variables into all the equations. This will help you check if the solution satisfies all the equations.

Q: What if one of the equations is not true?

A: If one of the equations is not true, then the solution is incorrect. You need to go back and check your work to see where you made a mistake.

Q: Can I use a calculator to verify a solution?

A: Yes, you can use a calculator to verify a solution. However, make sure to check your work by hand to ensure that the calculator is giving you the correct answer.

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

A: If you have a system of three or more equations, you can use the same method to verify the solution. However, you may need to use a more advanced method, such as substitution or elimination, to solve the system.

Q: Can I use a graphing calculator to verify a solution?

A: Yes, you can use a graphing calculator to verify a solution. Graphing calculators can help you visualize the solution and check if it satisfies all the equations.

Q: What if I'm not sure if the solution is correct?

A: If you're not sure if the solution is correct, you can try using a different method to solve the system. You can also ask a teacher or tutor for help.

Q: Can I use technology to verify a solution?

A: Yes, you can use technology, such as computer software or online tools, to verify a solution. These tools can help you check your work and ensure that the solution is correct.

Q: What are some common mistakes to avoid when verifying a solution?

A: Some common mistakes to avoid when verifying a solution include:

• Not substituting the values of the variables into all the equations
• Not checking the work by hand
• Not using a calculator or graphing calculator to verify the solution
• Not asking for help if you're unsure about the solution

Conclusion

Verifying a solution to a system of equations is an important step in solving systems of equations. By following the steps outlined in this article, you can ensure that the solution is correct and accurate. Remember to always check your work by hand and use technology, such as calculators or graphing calculators, to verify the solution.

Tips and Tricks

Here are some tips and tricks to help you verify solutions to systems of equations:

• Always substitute the values of the variables into all the equations
• Check the work by hand to ensure that the calculator is giving you the correct answer
• Use a graphing calculator to visualize the solution and check if it satisfies all the equations
• Ask a teacher or tutor for help if you're unsure about the solution
• Use technology, such as computer software or online tools, to verify the solution

Example Problems

Here are some example problems to try:

1. Verify the solution x = 2 and y = 3 to the system of equations:

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

2. Verify the solution x = -1 and y = 2 to the system of equations:

${ \begin{cases} x + 2y = 7 \\ 2x - 3y = 1 \end{cases} }$

Solutions

1. To verify the solution x = 2 and y = 3, substitute the values into both equations:

Equation 1: 2x + y = 5

Substituting x = 2 and y = 3 into the first equation, we get:

${ 2(2) + 3 = 5 }$

${ 4 + 3 = 5 }$

${ 7 = 5 }$

This equation is not true.

Equation 2: x - 2y = -3

Substituting x = 2 and y = 3 into the second equation, we get:

${ 2 - 2(3) = -3 }$

${ 2 - 6 = -3 }$

${ -4 = -3 }$

This equation is not true.

Since both equations are not true when x = 2 and y = 3, we can conclude that the given solution is incorrect.

2. To verify the solution x = -1 and y = 2, substitute the values into both equations:

Equation 1: x + 2y = 7

Substituting x = -1 and y = 2 into the first equation, we get:

${ (-1) + 2(2) = 7 }$

${ -1 + 4 = 7 }$

${ 3 = 7 }$

This equation is not true.

Equation 2: 2x - 3y = 1

Substituting x = -1 and y = 2 into the second equation, we get:

${ 2(-1) - 3(2) = 1 }$

${ -2 - 6 = 1 }$

${ -8 = 1 }$

This equation is not true.

Since both equations are not true when x = -1 and y = 2, we can conclude that the given solution is incorrect.