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Part 1
Solve the following systems of equations by​ elimination, if a solution exists.
left brace Start 2 By 1 Matrix 1st Row 1st Column 6 x plus 18 y equals 24 2nd Row 1st Column 15 y minus 20 equals negative 5 x EndMatrix

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Part 1
Select the correct choice below and fill in any answer boxes in your choice.
A.
The solution to the system is xequals
  
enter your response here​, yequals
  
enter your response here.
​(Type integers or simplified​ fractions.)
B.
There are infinitely many solutions.
C.
There is no solution.

1 answer

To solve the given system of equations by elimination, we first rewrite the equations clearly. The system can be expressed as:

  1. \( 6x + 18y = 24 \)
  2. \( 15y - 20 = -5x \)

We can rearrange the second equation for clarity:

\[ 5x + 15y = 20 \]

Now we have the system:

  1. \( 6x + 18y = 24 \)
  2. \( 5x + 15y = 20 \)

Next, we can use elimination to solve the equations. We will make the coefficients of \( x \) or \( y \) the same in both equations. Let's eliminate \( x \).

To do this, we can multiply the first equation by 5 and the second equation by 6:

\[ 5(6x + 18y) = 5(24) \implies 30x + 90y = 120 \] \[ 6(5x + 15y) = 6(20) \implies 30x + 90y = 120 \]

Now we have:

  1. \( 30x + 90y = 120 \)
  2. \( 30x + 90y = 120 \)

Since both equations are identical, it means that they represent the same line. Therefore, there are infinitely many solutions along that line.

So the answer is:

B. There are infinitely many solutions.