Select the correct answer.

The elimination method is ideal for solving this system of equations. By which number must you multiply the second equation to eliminate the [tex]$y$[/tex]-variable, and what is the solution for this system?

[tex]
\begin{array}{l}
x + 3y = 42 \\
2x - y = 14
\end{array}
[/tex]

A. Multiply the second equation by -3. The solution is [tex]$x=12, y=9$[/tex]

B. Multiply the second equation by -2. The solution is [tex][tex]$x=12, y=10$[/tex][/tex]

C. Multiply the second equation by 2. The solution is [tex]$x=15, y=9$[/tex]

D. Multiply the second equation by 3. The solution is [tex]$x=12, y=10$[/tex]



Answer :

To solve the system of equations
[tex]\[ \begin{array}{l} x + 3y = 42 \\ 2x - y = 14 \end{array} \][/tex]
using the elimination method, follow these steps:

1. Determine the Multiplication Factor:
To eliminate the [tex]\( y \)[/tex]-variable, the coefficients of [tex]\( y \)[/tex] in both equations must be made equal and opposite in value. If we multiply the second equation by 3, the coefficient of [tex]\( y \)[/tex] in the second equation will become -3, which is the negative of the coefficient of [tex]\( y \)[/tex] in the first equation.
[tex]\[ 3 \times (2x - y) = 3 \times 14 \][/tex]
This gives:
[tex]\[ 6x - 3y = 42 \][/tex]

2. Rewrite the System:
Now, the system of equations is:
[tex]\[ \begin{array}{l} x + 3y = 42 \\ 6x - 3y = 42 \end{array} \][/tex]

3. Add the Equations:
Adding these two equations will eliminate the [tex]\( y \)[/tex]-variable:
[tex]\[ (x + 3y) + (6x - 3y) = 42 + 42 \][/tex]
Simplifying, we get:
[tex]\[ 7x = 84 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{84}{7} = 12 \][/tex]

5. Substitute [tex]\( x \)[/tex] into the First Equation:
Substitute [tex]\( x = 12 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ 12 + 3y = 42 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ 3y = 42 - 12 \][/tex]
[tex]\[ 3y = 30 \][/tex]
[tex]\[ y = \frac{30}{3} = 10 \][/tex]

Therefore, the solution to the system is [tex]\( x = 12 \)[/tex] and [tex]\( y = 10 \)[/tex]. The correct choice is:

D. Multiply the second equation by 3. The solution is [tex]\( x = 12 \)[/tex], [tex]\( y = 10 \)[/tex].