Question

suppose you are solving the system of the following equations.\\(\

$$\begin{cases}6x + 6y = -5\\\\7x - 3y = 6\\end{cases}$$

\\)\
you decide to use the addition method by multiplying both sides of the second equation by 2. in which of the following was the multiplication performed correctly? explain.\
\
in which of the following was the multiplication performed correctly? explain. choose the correct answer below.\
\
\\(\bigcirc\\) a. \\(14x - 6y = 6\\)\
the side containing the variables is multiplied by the same nonzero number. adding the equations will eliminate the \\(y\\) variable.\
\\(\bigcirc\\) b. \\(14x - 6y = 12\\)\
both sides of the equation are multiplied by the same nonzero number. adding the equations will eliminate the \\(y\\) variable.

Explanation:

Step1: Recall the multiplication property of equality

When multiplying an equation by a number, we must multiply every term on both sides by that number. The second equation is \(7x - 3y=6\). We multiply both sides by 2.

Step2: Multiply each term of the second equation by 2

For the left - hand side: \(2\times(7x - 3y)=2\times7x-2\times3y = 14x-6y\)
For the right - hand side: \(2\times6 = 12\)
So the equation after multiplying both sides by 2 is \(14x - 6y = 12\). Also, when we add this new equation \(14x-6y = 12\) to the first equation \(6x + 6y=-5\), the \(y\) - terms will be eliminated (\((-6y)+6y = 0\)).

Step3: Analyze option A

In option A, the right - hand side is still 6. But we know that when we multiply the equation \(7x - 3y = 6\) by 2, the right - hand side should be \(2\times6=12\), not 6. So option A is incorrect.

Step4: Analyze option B

In option B, the equation is \(14x - 6y = 12\), which is obtained by multiplying each term of \(7x - 3y = 6\) by 2. And when we add this equation to \(6x + 6y=-5\), the \(y\) - variable will be eliminated. So option B is correct.

Answer:

B. \(14x - 6y = 12\)
Both sides of the equation are multiplied by the same nonzero number. Adding the equations will eliminate the \(y\) variable.