Certainly! Let's solve the system of equations step-by-step to determine if it has one solution, no solution, or infinitely many solutions. If there is one solution, we will identify the ordered pair [tex]\((x, y)\)[/tex].
The given system of equations is:
[tex]\[
\left\{\begin{array}{l}
y = -\frac{5}{3} x + 3 \\
y = \frac{1}{3} x - 3
\end{array}\right.
\][/tex]
### Step 1: Set the Equations Equal to Each Other
Since both equations are equal to [tex]\(y\)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
-\frac{5}{3} x + 3 = \frac{1}{3} x - 3
\][/tex]
### Step 2: Solve for [tex]\(x\)[/tex]
Next, we solve for [tex]\(x\)[/tex]. First, we eliminate the fractions by multiplying every term by 3 to simplify the equation:
[tex]\[
-5x + 9 = x - 9
\][/tex]
Now, we move all the [tex]\(x\)[/tex]-terms to one side and the constant terms to the other side:
[tex]\[
-5x - x = -9 - 9
\][/tex]
[tex]\[
-6x = -18
\][/tex]
Divide both sides by -6 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3
\][/tex]
### Step 3: Solve for [tex]\(y\)[/tex]
Now that we have [tex]\(x = 3\)[/tex], we can substitute this value back into either of the original equations to find [tex]\(y\)[/tex]. We'll use the first equation:
[tex]\[
y = -\frac{5}{3} (3) + 3
\][/tex]
Simplify:
[tex]\[
y = -5 + 3
\][/tex]
[tex]\[
y = -2
\][/tex]
### Conclusion
The solution to the system of equations is the ordered pair [tex]\((x, y) = (3, -2)\)[/tex]. Therefore, the system has one solution, which is:
[tex]\[
\boxed{(3, -2)}
\][/tex]
