Determine the number of solutions of this system of linear equations:x+2y=−44x+8y=−16This question requires you to show your work.(1 point)Responses One Solution One Solution Infinitely Many Solutions Infinitely Many Solutions No Solutions

Answer 1

To determine the number of solutions for the given system of linear equations, we need to analyze the equations:

1. \( x + 2y = -44 \) (Equation 1)
2. \( -x + 8y = -16 \) (Equation 2)

Let's manipulate the equations step by step.

Step 1: Rearranging Equation 2

We can rearrange Equation 2 to a more standard form:

\[ -x + 8y = -16 \implies x = 8y + 16 \]

Step 2: Substitute Equation 2 into Equation 1

We can substitute \( x \) from Equation 2 into Equation 1:

\[ (8y + 16) + 2y = -44 \]

Step 3: Combine the terms

Combine like terms:

\[ 8y + 2y + 16 = -44 \implies 10y + 16 = -44 \]

Step 4: Solve for y

Now, isolate \( y \):

\[ 10y = -44 - 16 \implies 10y = -60 \implies y = -6 \]

Step 5: Find x using y

Now that we have \( y = -6 \), we can find \( x \) using the expression for \( x \) derived from Equation 2:

\[ x = 8(-6) + 16 = -48 + 16 = -32 \]

Conclusion

We found \( y = -6 \) and \( x = -32 \). This indicates that there is one unique solution to the system of equations:

\[ (x, y) = (-32, -6) \]

Thus, the answer is:

One Solution