Determine If The Ordered Pair Provided Is A Solution To The Linear System.${ \begin{align*} x + 8y &= 43 \ 3x - 2y &= -1 \end{align*} }$Given The Ordered Pair { (3, 5)$}$.A. Yes, { (3, 5)$}$ Is A Solution To The

Introduction

In mathematics, a linear system is a set of linear equations that can be solved using various methods. One of the most common methods is the substitution method, where we substitute the value of one variable from one equation into the other equation to solve for the remaining variable. In this article, we will determine if the ordered pair (3, 5) is a solution to the linear system given by the equations:

$$\begin{align*} x + 8y &= 43 \\ 3x - 2y &= -1 \end{align*}$$

Understanding the Linear System

A linear system is a set of linear equations that can be written in the form:

$$\begin{align*} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{align*}$$

where $a_1, b_1, c_1, a_2, b_2,$ and $c_2$ are constants, and $x$ and $y$ are variables. In this case, we have two linear equations with two variables, $x$ and $y$.

The Substitution Method

The substitution method is a common method used to solve linear systems. The idea is to substitute the value of one variable from one equation into the other equation to solve for the remaining variable. In this case, we can substitute the value of $x$ from the first equation into the second equation to solve for $y$.

Step 1: Substitute the Value of x from the First Equation into the Second Equation

We can substitute the value of $x$ from the first equation into the second equation as follows:

$$\begin{align*} 3x - 2y &= -1 \\ 3(43 - 8y) - 2y &= -1 \end{align*}$$

Simplifying the equation, we get:

$$\begin{align*} 129 - 24y - 2y &= -1 \\ -26y &= -130 \\ y &= 5 \end{align*}$$

Step 2: Substitute the Value of y into the First Equation to Solve for x

Now that we have the value of $y$, we can substitute it into the first equation to solve for $x$.

$$\begin{align*} x + 8y &= 43 \\ x + 8(5) &= 43 \\ x + 40 &= 43 \\ x &= 3 \end{align*}$$

Conclusion

In conclusion, we have determined that the ordered pair (3, 5) is a solution to the linear system given by the equations:

$$\begin{align*} x + 8y &= 43 \\ 3x - 2y &= -1 \end{align*}$$

The substitution method was used to solve the linear system, and we found that the ordered pair (3, 5) satisfies both equations.

Final Answer

The final answer is: Yes, (3, 5) is a solution to the linear system.

Additional Resources

For more information on linear systems and the substitution method, please refer to the following resources:

• Linear Systems
• Substitution Method

Related Topics

• Linear Equations
• Systems of Linear Equations
• Substitution Method
  Determine if the Ordered Pair Provided is a Solution to the Linear System: Q&A ====================================================================

Introduction

In our previous article, we determined that the ordered pair (3, 5) is a solution to the linear system given by the equations:

$$\begin{align*} x + 8y &= 43 \\ 3x - 2y &= -1 \end{align*}$$

In this article, we will answer some frequently asked questions (FAQs) related to linear systems and the substitution method.

Q&A

Q: What is a linear system?

A: A linear system is a set of linear equations that can be written in the form:

$$\begin{align*} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{align*}$$

where $a_1, b_1, c_1, a_2, b_2,$ and $c_2$ are constants, and $x$ and $y$ are variables.

Q: What is the substitution method?

A: The substitution method is a common method used to solve linear systems. The idea is to substitute the value of one variable from one equation into the other equation to solve for the remaining variable.

Q: How do I determine if an ordered pair is a solution to a linear system?

A: To determine if an ordered pair is a solution to a linear system, you need to substitute the values of the variables from the ordered pair into both equations and check if the resulting equations are true.

Q: What if the ordered pair does not satisfy both equations?

A: If the ordered pair does not satisfy both equations, then it is not a solution to the linear system.

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

A: Yes, you can use the substitution method to solve a linear system with more than two variables. However, you need to be careful when substituting the values of the variables, as it can become complicated.

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

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

• Not checking if the ordered pair satisfies both equations
• Not simplifying the equations correctly
• Not substituting the values of the variables correctly

Q: Can I use the substitution method to solve a linear system with no solution?

A: Yes, you can use the substitution method to solve a linear system with no solution. However, if the ordered pair does not satisfy both equations, then it is not a solution to the linear system.

Conclusion

In conclusion, we have answered some frequently asked questions related to linear systems and the substitution method. We hope that this article has provided you with a better understanding of how to determine if an ordered pair is a solution to a linear system.

Final Answer

The final answer is: Yes, (3, 5) is a solution to the linear system.

Additional Resources

For more information on linear systems and the substitution method, please refer to the following resources:

• Linear Systems
• Substitution Method

Related Topics

• Linear Equations
• Systems of Linear Equations
• Substitution Method