Solve the following systems of equations by​ elimination, if a solution exists.

Question

left brace Start 2 By 1 Matrix 1st Row 1st Column 4 x equals 12 minus 8 y 2nd Row 1st Column 5 x plus 3 y equals 6 EndMatrix

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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box within 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.

Answer 1

To solve the given system of equations using elimination, we first rewrite the equations in standard form.

The given equations are:

1. \( 4x = 12 - 8y \)
2. \( 5x + 3y = 6 \)

We can rearrange the first equation to standard form:

\[ 4x + 8y = 12 \]

Now we have the system of equations:

1. \( 4x + 8y = 12 \)
2. \( 5x + 3y = 6 \)

Next, we can eliminate one of the variables. Let's eliminate \( y \) by first making the coefficients of \( y \) in both equations equal.

To do this, we can multiply the entire first equation by 3 and the entire second equation by 8:

1. \( 3(4x + 8y) = 3(12) \)

   • Which simplifies to: \( 12x + 24y = 36 \)

2. \( 8(5x + 3y) = 8(6) \)

   • Which simplifies to: \( 40x + 24y = 48 \)

Now, we have a new system:

1. \( 12x + 24y = 36 \)
2. \( 40x + 24y = 48 \)

Next, we subtract the first equation from the second equation:

\[ (40x + 24y) - (12x + 24y) = 48 - 36 \]

This simplifies to:

\[ 28x = 12 \]

Now, solve for \( x \):

\[ x = \frac{12}{28} = \frac{3}{7} \]

Now that we have \( x \), we can substitute \( x \) back into one of the original equations to find \( y \). We'll use the first equation:

\[ 4x + 8y = 12 \]

Substituting \( x = \frac{3}{7} \):

\[ 4\left(\frac{3}{7}\right) + 8y = 12 \]

This simplifies to:

\[ \frac{12}{7} + 8y = 12 \]

Subtract \(\frac{12}{7}\) from both sides:

\[ 8y = 12 - \frac{12}{7} \]

To combine the terms on the right, convert 12 into a fraction with a denominator of 7:

\[ 12 = \frac{84}{7} \]

So:

\[ 8y = \frac{84}{7} - \frac{12}{7} = \frac{72}{7} \]

Now, divide both sides by 8:

\[ y = \frac{72}{7} \cdot \frac{1}{8} = \frac{72}{56} = \frac{9}{7} \]

Thus, the solution to the system of equations is:

\[ x = \frac{3}{7}, \quad y = \frac{9}{7} \]

Conclusion

The correct choice is:

A. The solution to the system is \( x = \frac{3}{7} \), \( y = \frac{9}{7} \).