Question

How many solutions does the system of equations have?
8x−6y=48

8x+6y=48
(1 point)
Responses

infinite
infinite

one
one

none
none

two

Answer 1

To determine the number of solutions for the given system of equations, we can analyze the equations directly. The system is:

1. \( 8x - 6y = 48 \) (Equation 1)
2. \( 8x + 6y = 48 \) (Equation 2)

We can solve this system by using the method of elimination or substitution. Let's add the two equations together:

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

This simplifies to:

\[
16x = 96
\]

Now, dividing both sides by 16:

\[
x = 6
\]

Now that we have a value for \(x\), we can substitute it back into either equation to find \(y\). Let's use Equation 1:

\[
8(6) - 6y = 48
\]

This simplifies to:

\[
48 - 6y = 48
\]

Subtracting 48 from both sides gives:

\[
-6y = 0
\]

Dividing by -6:

\[
y = 0
\]

We found a unique solution: \( (6, 0) \).

Since we have found a specific value for both variables, the system has exactly **one** solution.

So, the answer is **one**.