Question

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

$$\begin{cases}x + 2y = 8\\\\4x + 8y = 32\\end{cases}$$

\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 = \square, y\text{ any real number}\\}\\). (simplify your answer. type an expression using \\(y\\) as the variable as needed.)\
\\(\bigcirc\\) c. the system is inconsistent.

Explanation:

Step1: Simplify the second equation

Divide the second equation \(4x + 8y = 32\) by 4. We get \(x + 2y = 8\), which is the same as the first equation \(x + 2y = 8\).

Step2: Analyze the system

Since both equations are identical, they represent the same line. So, there are infinitely many solutions. We can express \(x\) from the first equation: \(x = 8 - 2y\). So the solutions are all ordered pairs \((x,y)\) where \(x = 8 - 2y\) and \(y\) is any real number.

Answer:

B. There are infinitely many solutions. Using ordered pairs, they can be expressed as \(\{(x,y)\mid x = 8 - 2y, y\text{ any real number}\}\)