For each system of equations, drag the true statement about its solution set to the box under the system.

[tex]\[
\begin{array}{l}
y=6x-3 \\
y=3(2x+1)
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
y=2x+1 \\
y=6x+3
\end{array}
\][/tex]

Statements to drag:

- Zero Solutions: [tex]$6x-3$[/tex] can never be equal to [tex]$6x+3$[/tex].
- Zero Solutions: [tex]$y$[/tex] is equal to two different expressions.
- One Solution: [tex]$2x+1=6x+3$[/tex] has one solution.
- Infinitely Many Solutions: Both equations are equal to [tex]$y$[/tex].
- Infinitely Many Solutions: [tex]$6x+3$[/tex] is a multiple of [tex]$2x+1$[/tex].



Answer :

Let's analyze each of the given systems of equations to determine their solution sets.

### System 1
[tex]\[ \begin{array}{l} y=6x-3 \\ y=3(2x+1) \end{array} \][/tex]

First, we simplify the second equation:
[tex]\[ 3(2x+1) = 6x + 3 \][/tex]

Thus, the system becomes:
[tex]\[ \begin{array}{l} y=6x-3 \\ y=6x+3 \end{array} \][/tex]

Let's set the right sides equal to each other:
[tex]\[ 6x-3 = 6x+3 \][/tex]

Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ -3 = 3 \][/tex]

This is a contradiction, which means there are no values of [tex]\(x\)[/tex] that satisfy both equations simultaneously. Therefore, this system has zero solutions.

### System 2
[tex]\[ \begin{array}{l} y=2x+1 \\ y=6x+3 \end{array} \][/tex]

Set the right sides equal to each other:
[tex]\[ 2x+1 = 6x+3 \][/tex]

Rearrange the terms to solve for [tex]\(x\)[/tex]:
[tex]\[ 2x - 6x = 3 - 1 \\ -4x = 2 \\ x = -\frac{1}{2} \][/tex]

Plug [tex]\(x = -\frac{1}{2}\)[/tex] back into either equation to find [tex]\(y\)[/tex]. Using the first equation:
[tex]\[ y = 2\left(-\frac{1}{2}\right) + 1 = -1 + 1 = 0 \][/tex]

So, the solution is [tex]\((- \frac{1}{2}, 0)\)[/tex]. Therefore, this system has one solution.

### Summary of Results

1. System 1:
[tex]\[ \begin{array}{l} y=6x-3 \\ y=3(2x+1) \end{array} \][/tex]
- Zero Solutions: [tex]\(6x-3\)[/tex] can never be equal to [tex]\(6x+3\)[/tex].

2. System 2:
[tex]\[ \begin{array}{l} y=2x+1 \\ y=6x+3 \end{array} \][/tex]
- One Solution: [tex]\(2x+1 = 6x+3\)[/tex] has one solution.

Therefore, we align the statements accordingly for the systems of equations:

1. Zero Solutions: [tex]\(6x-3\)[/tex] can never be equal to [tex]\(6x+3\)[/tex].
- System:
[tex]\[ \begin{array}{l} y=6x-3 \\ y=3(2x+1) \end{array} \][/tex]

2. One Solution: [tex]\(2x+1 = 6x+3\)[/tex] has one solution.
- System:
[tex]\[ \begin{array}{l} y=2x+1 \\ y=6x+3 \end{array} \][/tex]