Question

How many solutions does the system of equations have?
8x−6y=48
8x+6y=48 (1 point)
Responses

two
two

infinite
infinite

one
one

none

Answer 1

To determine how many solutions the system of equations has, we can analyze the two equations:

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

We can solve the system of equations by manipulation. Let's start by adding the two equations:

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

This simplifies to:

\[
16x = 96
\]

Now, we can solve for \(x\):

\[
x = \frac{96}{16} = 6
\]

Next, we can substitute \(x = 6\) back into one of the original equations to find \(y\). Let's use the first equation:

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

Simplifying that gives:

\[
48 - 6y = 48
\]

Subtracting 48 from both sides results in:

\[
-6y = 0
\]

Thus:

\[
y = 0
\]

So we have found a unique solution:

\[
(x, y) = (6, 0)
\]

Now, we can check the second equation with these values:

\[
8(6) + 6(0) = 48
\]

This simplifies to \(48 = 48\), confirming that the solution is valid.

Since we have one unique solution for \(x\) and \(y\), the system of equations has:

**one** solution.