Sure, I can explain step-by-step how we determine if the system of equations has infinite solutions.
We are given two equations:
[tex]\[
\begin{cases}
2x + y = 3 \\
6x = 9 - 3y
\end{cases}
\][/tex]
Step 1: Transform the second equation to match the form of the first equation.
[tex]\[
6x = 9 - 3y
\][/tex]
We can rewrite this as:
[tex]\[
6x + 3y = 9
\][/tex]
Step 2: Simplify the second equation.
Notice that both the first term on the left side and the constant term on the right side are multiples of 3. Thus, let's divide the entire equation by 3:
[tex]\[
\frac{6x}{3} + \frac{3y}{3} = \frac{9}{3}
\][/tex]
Simplifying, we get:
[tex]\[
2x + y = 3
\][/tex]
Step 3: Compare the two equations.
[tex]\[
\begin{cases}
2x + y = 3 \\
2x + y = 3
\end{cases}
\][/tex]
Step 4: Observation.
We see that both equations are identical. When the two equations are identical, it means that every solution of one equation is also a solution of the other. This implies that there are infinitely many solutions because any value of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfies the first equation will also satisfy the second equation.
Conclusion:
Since the two equations are the same, they represent the same line in the coordinate plane, and there are infinitely many points (solutions) that lie on this line.
Therefore, the system of equations has infinite solutions.
