Alright, let's solve the system of linear equations step-by-step:
Given the system of equations:
[tex]\[
3x - y = 3 \quad \text{(1)}
\][/tex]
[tex]\[
9x - 3y = 9 \quad \text{(2)}
\][/tex]
First, notice that equation (2) is a multiple of equation (1). We can see this by dividing equation (2) by 3:
[tex]\[
\frac{9x - 3y}{3} = \frac{9}{3}
\][/tex]
This simplifies to:
[tex]\[
3x - y = 3 \quad \text{(3)}
\][/tex]
Now, we see that equation (3) is actually the same as equation (1). This indicates that both equations represent the same line, meaning they are dependent. Because the two equations are essentially identical, every point on the line [tex]\(3x - y = 3\)[/tex] is a solution to the system.
Thus, we can express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex] using equation (1). Solving for [tex]\(y\)[/tex]:
[tex]\[
3x - y = 3
\][/tex]
[tex]\[
y = 3x - 3
\][/tex]
Therefore, the solutions to the system of equations can be written in the form:
[tex]\[
(x, y) = (x, 3x-3)
\][/tex]
where [tex]\(x\)[/tex] can be any real number. This represents an infinite number of solutions along the line defined by the equation [tex]\(3x - y = 3\)[/tex].
