Solve the system by the method of your choice.

left brace Start 2 By 1 Matrix 1st Row 1st Column x equals 8 y plus 5 2nd Row 1st Column negative 2 x plus 16 y equals negative 10 EndMatrix

x=8y+5

−2x+16y=−10

Question content area bottom

Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

There is exactly one solution. The solution set is 

StartSet nothing EndSetenter your response here. 

​(Simplify your answer. Type an ordered​ pair.)

B.

There are infinitely many solutions. The solution set is 

StartSet left parenthesis x comma y right parenthesis vertical line x equals 8 y plus 5 EndSet{(x,y) x=8y+5} 

or 

StartSet left parenthesis x comma y right parenthesis vertical line minus 2 x plus 16 y equals negative 10 EndSet{(x,y) −2x+16y=−10}.

C.

The solution set is 

empty set∅.

To solve the system of equations given by

\[
\begin{cases}
x = 8y + 5 \\
-2x + 16y = -10
\end{cases}
\]

we can use substitution since the first equation directly gives us \( x \) in terms of \( y \).

First, let's substitute \( x = 8y + 5 \) into the second equation:

\[
-2(8y + 5) + 16y = -10
\]

Distribute \(-2\) in the equation:

\[
-16y - 10 + 16y = -10
\]

Combine like terms:

\[
-10 = -10
\]

Since this equation is always true, it indicates that the system has infinitely many solutions. The equations are dependent, meaning one equation is a multiple of the other, and any value of \( y \) that satisfies one equation will satisfy the other.

Therefore, the solution set is all ordered pairs \((x, y)\) that satisfy the equation \( x = 8y + 5 \).

So, the correct choice is:

**B. There are infinitely many solutions. The solution set is \( \{(x, y) \mid x = 8y + 5\} \).**

Do you want details or have any questions?

Here are 5 questions you can ask next:
1. How do you solve a system of equations using the elimination method?
2. Can you show how to graphically represent the solution of a system of equations?
3. How do you determine if a system of equations has no solution?
4. What are the steps to solve a system of linear equations using matrices?
5. Can you explain how to solve a system of equations using Gaussian elimination?

**Tip:** When dealing with systems of equations, always check for consistency and dependency to determine if the system has a unique solution, infinitely many solutions, or no solution.

Learn how to solve a system of equations using the substitution method. This problem involves two linear equations, where one equation is dependent on the other. Explore the concept of dependent systems and understand how to find infinitely many solutions.

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