To solve the systems of equations, we need to compute each one step-by-step to determine their solutions or determine if there are no solutions or an infinite number of solutions.
### System 1
[tex]\[
\begin{cases}
2x + 6y = 4 \\
x = 4y + 2
\end{cases}
\][/tex]
Step 1: Substitute [tex]\( x = 4y + 2 \)[/tex] into [tex]\( 2x + 6y = 4 \)[/tex]:
[tex]\[
2(4y + 2) + 6y = 4
\][/tex]
[tex]\[
8y + 4 + 6y = 4
\][/tex]
[tex]\[
14y + 4 = 4
\][/tex]
[tex]\[
14y = 0
\][/tex]
[tex]\[
y = 0
\][/tex]
Step 2: Substitute [tex]\( y = 0 \)[/tex] back into [tex]\( x = 4y + 2 \)[/tex]:
[tex]\[
x = 4(0) + 2 = 2
\][/tex]
The solution is [tex]\( (2, 0) \)[/tex], which is not one of our matching responses. Therefore, it does not match any of the provided options.
### System 2
[tex]\[
\begin{cases}
-3x - 4y = -6 \\
x = -2 - \frac{4}{3}y
\end{cases}
\][/tex]
Step 1: Substitute [tex]\( x = -2 - \frac{4}{3}y \)[/tex] into [tex]\( -3x - 4y = -6 \)[/tex]:
[tex]\[
-3(-2 - \frac{4}{3}y) - 4y = -6
\][/tex]
[tex]\[
6 + 4y - 4y = -6
\][/tex]
[tex]\[
6 = -6
\][/tex]
Since [tex]\( 6 \neq -6 \)[/tex], this system is inconsistent and has no solutions.
### System 3
[tex]\[
\begin{cases}
3x - 4y = 6 \\
x = \frac{4}{3}y + 2
\end{cases}
\][/tex]
Step 1: Substitute [tex]\( x = \frac{4}{3}y + 2 \)[/tex] into [tex]\( 3x - 4y = 6 \)[/tex]:
[tex]\[
3(\frac{4}{3}y + 2) - 4y = 6
\][/tex]
[tex]\[
4y + 6 - 4y = 6
\][/tex]
[tex]\[
6 = 6
\][/tex]
Since the equation simplifies to an identity (true for all [tex]\( y \)[/tex]), this system has an infinite number of solutions.
### Matching
- The first system [tex]\( 2x + 6y = 4 \)[/tex] and [tex]\( x = 4y + 2 \)[/tex] has the solution [tex]\( (2, 0) \)[/tex], which does not match the provided options.
- The second system [tex]\( -3x - 4y = -6 \)[/tex] and [tex]\( x = -2 - \frac{4}{3} y \)[/tex] has no solution.
- The third system [tex]\( 3x - 4y = 6 \)[/tex] and [tex]\( x = \frac{4}{3} y + 2 \)[/tex] has an infinite number of solutions.
Providing the correct matches:
- The system
[tex]\[
\begin{cases}
2x + 6y = 4 \\
x = 4y + 2
\end{cases}
\][/tex]
does not match any of the provided answers.
- The system
[tex]\[
\begin{cases}
-3x - 4y = -6 \\
x = -2 - \frac{4}{3} y
\end{cases}
\][/tex]
has no solution.
- The system
[tex]\[
\begin{cases}
3x - 4y = 6 \\
x = \frac{4}{3} y + 2
\end{cases}
\][/tex]
has an infinite number of solutions.
