Solve the system of equations by any convenient​ method, if a solution exists.

Question

left brace Start 2 By 1 Matrix 1st Row 1st Column 4 x plus 3 y equals 0 2nd Row 1st Column 8 x plus 6 y equals 7 EndMatrix

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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution is xequals
  
enter your response here​, yequals
  
enter your response here.
​(Simplify your​ answers.)
B.
There are infinitely many solutions.
C.
There is no solution.

Answer 1

To solve the system of equations

\[ 4x + 3y = 0 \tag{1} \] \[ 8x + 6y = 7 \tag{2} \]

we can start by manipulating one of the equations to compare them.

Notice that the second equation can be simplified:

1. Divide the entire equation (2) by 2:

\[ 4x + 3y = 3.5 \tag{3} \]

Now we have two equations:

\[ 4x + 3y = 0 \tag{1} \]

\[ 4x + 3y = 3.5 \tag{3} \]

Now we can see that both equations share the same left-hand side, which implies they are equivalent only if their right-hand sides are equal. However, they have different right-hand sides (0 and 3.5 respectively).

This indicates that the two lines represented by the equations are parallel and do not intersect. Thus, there are no solutions to the system of equations.

The correct answer is C. There is no solution.