To solve the given system of equations:
[tex]\[
\begin{cases}
3x - y = 3 \\
9x - 3y = 9
\end{cases}
\][/tex]
we can proceed with a step-by-step approach to find a solution.
### Step 1: Analyze the equations
The two equations we have are:
1. [tex]\( 3x - y = 3 \)[/tex]
2. [tex]\( 9x - 3y = 9 \)[/tex]
### Step 2: Simplify the second equation
Notice that the second equation appears to be a multiple of the first equation. Let's simplify the second equation:
[tex]\[
9x - 3y = 9
\][/tex]
Divide the entire equation by 3:
[tex]\[
\frac{9x}{3} - \frac{3y}{3} = \frac{9}{3}
\][/tex]
Simplifying, we get:
[tex]\[
3x - y = 3
\][/tex]
### Step 3: Compare the equations
Now we see that both equations are the same:
[tex]\[
3x - y = 3
\][/tex]
This indicates that every solution that satisfies one equation will satisfy the other. This means we have infinitely many solutions along a line described by this equation.
### Step 4: Express y in terms of x
From the equation [tex]\( 3x - y = 3 \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[
3x - y = 3
\][/tex]
Add [tex]\( y \)[/tex] to both sides of the equation:
[tex]\[
3x = y + 3
\][/tex]
Subtract 3 from both sides:
[tex]\[
3x - 3 = y
\][/tex]
Thus, we can write:
[tex]\[
y = 3x - 3
\][/tex]
### Step 5: Express in terms of y
If you prefer, we can express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Starting again from [tex]\( 3x - y = 3 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[
3x = y + 3
\][/tex]
Divide by 3:
[tex]\[
x = \frac{y + 3}{3}
\][/tex]
So, the solution set can also be expressed as:
[tex]\[
x = \frac{y}{3} + 1
\][/tex]
### Conclusion
Therefore, the solutions to the system of equations are:
[tex]\[
\boxed{x = \frac{y}{3} + 1}
\][/tex]
This represents the infinite number of solutions where [tex]\( x \)[/tex] and [tex]\( y \)[/tex] satisfy the relationship [tex]\( x = \frac{y}{3} + 1 \)[/tex].
