Use The Substitution Method To Solve The System Of Equations. Choose The Correct Ordered Pair.${ \begin{array}{l} 2x + Y = 8 \ y = -x + 5 \end{array} }$A. { (4,1)$}$ B. { (2,3)$}$ C. { (5,0)$}$ D.

Mar 13, 2025 by ADMIN 201 views

**Solving Systems of Equations Using the Substitution Method**

Introduction

Solving systems of equations is a fundamental concept in algebra, and one of the most effective methods for solving these systems is the substitution method. This method involves substituting one equation into the other to solve for the variables. In this article, we will explore how to use the substitution method to solve a system of equations, and we will apply this method to a specific system of equations to find the correct ordered pair.

The Substitution Method

The substitution method is a step-by-step process that involves substituting one equation into the other to solve for the variables. Here are the steps to follow:

Identify the equations: Identify the two equations in the system and determine which one will be substituted into the other.
Solve for one variable: Solve one of the equations for one of the variables. This will give you an expression that can be substituted into the other equation.
Substitute the expression: Substitute the expression from step 2 into the other equation.
Solve for the other variable: Solve the resulting equation for the other variable.
Check the solution: Check the solution by plugging it back into both original equations.

Applying the Substitution Method to a System of Equations

Let's apply the substitution method to the following system of equations:

{ \begin{array}{l} 2x + y = 8 \\ y = -x + 5 \end{array} \}

To solve this system, we will use the substitution method. We will substitute the expression for y from the second equation into the first equation.

Step 1: Identify the equations

The two equations in the system are:

$2x + y = 8$
$y = -x + 5$

Step 2: Solve for one variable

We will solve the second equation for y:

$y = -x + 5$

Step 3: Substitute the expression

We will substitute the expression for y into the first equation:

$2x + (-x + 5) = 8$

Step 4: Solve for the other variable

We will simplify the equation and solve for x:

$2x - x + 5 = 8$

$ x + 5 = 8$

$ x = 3$

Step 5: Check the solution

We will plug the value of x back into both original equations to check the solution:

$2x + y = 8$

$2(3) + y = 8$

$6 + y = 8$

$y = 2$

$y = -x + 5$

$y = -(3) + 5$

$y = 2$

The solution checks out, and we have found the correct ordered pair.

Conclusion

In this article, we have explored how to use the substitution method to solve a system of equations. We have applied this method to a specific system of equations and found the correct ordered pair. The substitution method is a powerful tool for solving systems of equations, and it is an essential concept in algebra.

Choosing the Correct Ordered Pair

Now that we have solved the system of equations, we need to choose the correct ordered pair from the options provided. The correct ordered pair is:

$\boxed{(3,2)}$

However, this is not one of the options provided. Let's re-examine the options and see if we can find the correct ordered pair.

Re-examining the Options

Let's re-examine the options and see if we can find the correct ordered pair.

A. $\boxed{(4,1)}$

B. $\boxed{(2,3)}$

C. $\boxed{(5,0)}$

D. $\boxed{(3,2)}$

We have already found the correct ordered pair to be (3,2), but this is not one of the options provided. However, we can see that option A is close to the correct ordered pair. Let's plug option A back into both original equations to see if it checks out.

$2x + y = 8$

$2(4) + 1 = 8$

$8 + 1 = 8$

$9 = 8$

This does not check out, so option A is not the correct ordered pair.

$y = -x + 5$

$y = -(4) + 5$

$y = 1$

This checks out, so option A is not the correct ordered pair, but it is close.

Let's try option B.

$2x + y = 8$

$2(2) + 3 = 8$

$4 + 3 = 8$

$7 = 8$

This does not check out, so option B is not the correct ordered pair.

$y = -x + 5$

$y = -(2) + 5$

$y = 3$

This checks out, so option B is not the correct ordered pair, but it is close.

Let's try option C.

$2x + y = 8$

$2(5) + 0 = 8$

$10 + 0 = 8$

$10 = 8$

This does not check out, so option C is not the correct ordered pair.

$y = -x + 5$

$y = -(5) + 5$

$y = 0$

This checks out, so option C is not the correct ordered pair, but it is close.

Let's try option D.

$2x + y = 8$

$2(3) + 2 = 8$

$6 + 2 = 8$

$8 = 8$

This checks out, so option D is the correct ordered pair.

The final answer is: $\boxed{D}$

Introduction

Solving systems of equations is a fundamental concept in algebra, and one of the most effective methods for solving these systems is the substitution method. In this article, we will answer some frequently asked questions (FAQs) about solving systems of equations using the substitution method.

Q: What is the substitution method?

A: The substitution method is a step-by-step process that involves substituting one equation into the other to solve for the variables. This method is used to solve systems of equations where one equation is already solved for one of the variables.

Q: How do I know which equation to substitute into the other?

A: To determine which equation to substitute into the other, look for the equation that is already solved for one of the variables. This equation will be substituted into the other equation to solve for the remaining variable.

Q: What if I have two equations with two variables, but neither equation is solved for one of the variables?

A: In this case, you can use the elimination method to solve the system of equations. The elimination method involves adding or subtracting the two equations to eliminate one of the variables.

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

A: No, the substitution method is only used to solve systems of equations with two variables. If you have a system of equations with more than two variables, you will need to use a different method, such as the elimination method or the graphing method.

Q: How do I check my solution to make sure it is correct?

A: To check your solution, plug the values of the variables back into both original equations to make sure they are true. If the solution checks out, then you have found the correct ordered pair.

Q: What if I get a solution that doesn't check out?

A: If you get a solution that doesn't check out, then you need to go back and recheck your work. Make sure you are following the correct steps and that you are substituting the correct expressions into the other equation.

Q: Can I use the substitution method to solve a system of equations with fractions or decimals?

A: Yes, you can use the substitution method to solve a system of equations with fractions or decimals. Just make sure to follow the correct steps and to simplify the expressions as you go.

Q: How do I know if I have found the correct ordered pair?

A: To determine if you have found the correct ordered pair, plug the values of the variables back into both original equations to make sure they are true. If the solution checks out, then you have found the correct ordered pair.

Conclusion

Solving systems of equations using the substitution method can be a powerful tool for solving algebraic equations. By following the correct steps and checking your solution, you can find the correct ordered pair and solve the system of equations. Remember to always check your work and to simplify the expressions as you go.

Additional Resources

If you are having trouble solving systems of equations using the substitution method, there are many additional resources available to help you. Some of these resources include:

Online tutorials and videos
Algebra textbooks and workbooks
Online practice problems and quizzes
Algebra software and calculators

By using these resources and practicing the substitution method, you can become more confident and proficient in solving systems of equations.

Final Tips

Always check your work and simplify the expressions as you go.
Use the substitution method to solve systems of equations with two variables.
Use the elimination method to solve systems of equations with more than two variables.
Practice, practice, practice! The more you practice, the more confident and proficient you will become in solving systems of equations.

By following these tips and using the substitution method, you can become a master of solving systems of equations and tackle even the most challenging algebraic equations.