Question

solve the system of equations. if the system has no solution, say that it is inconsistent.
\

$$\begin{cases} 2x - 6y = 4 \\\\ 10x + 6y = 26 \\end{cases}$$

select the correct choice below and, if necessary, fill in any answer boxes within your choice.

\bigcirc a. the solution is x = \square and y = \square. (type integers or simplified fractions.)

\bigcirc b. there are infinitely many solutions. using ordered pairs, they can be expressed as \\{(x,y) \mid x =
(simplify your answer. type an expression using y as the variable as needed.)

\bigcirc c. the system is inconsistent.

Explanation:

Step1: Add the two equations

We have the system:

$$
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$$

Adding the two equations to eliminate \(y\):
\((2x - 6y)+(10x + 6y)=4 + 26\)
Simplify the left - hand side and the right - hand side:
\(2x+10x-6y + 6y=30\)
\(12x=30\)

Step2: Solve for \(x\)

Divide both sides of the equation \(12x = 30\) by \(12\):
\(x=\frac{30}{12}=\frac{5}{2}\)

Step3: Substitute \(x = \frac{5}{2}\) into one of the original equations

Let's substitute \(x=\frac{5}{2}\) into the first equation \(2x-6y = 4\).
\(2\times\frac{5}{2}-6y=4\)
Simplify the left - hand side: \(5-6y = 4\)

Step4: Solve for \(y\)

Subtract \(5\) from both sides of the equation \(5-6y = 4\):
\(-6y=4 - 5=-1\)
Divide both sides by \(-6\):
\(y=\frac{-1}{-6}=\frac{1}{6}\)

Answer:

A. The solution is \(x=\frac{5}{2}\) and \(y=\frac{1}{6}\)