Question

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

$$\begin{cases} 2x + y = 4 \\\\ 4x + 2y = 3 \\end{cases}$$

select the correct choice below and, if necessary, fill in any answer boxes within your choice. \
a. the solution is x = \square and y = \square. (type integers or simplified fractions.) \
b. there are infinitely many solutions. using ordered pairs, they can be expressed as \\{(x,y) \mid x = \square, y is any real number\\}. (simplify your answer. type an expression using y as the variable as needed.) \
c. the system is inconsistent. \
use the graphing tool to graph the lines. \
click to enlarge graph

Explanation:

Step1: Rewrite equations in slope - intercept form

The first equation is \(2x + y=4\). Solve for \(y\):
\(y=- 2x + 4\). The slope \(m_1=-2\) and the \(y\) - intercept \(b_1 = 4\).
The second equation is \(4x+2y = 3\). Solve for \(y\):
First, subtract \(4x\) from both sides: \(2y=-4x + 3\). Then divide by \(2\): \(y=-2x+\frac{3}{2}\). The slope \(m_2=-2\) and the \(y\) - intercept \(b_2=\frac{3}{2}\).

Step2: Analyze the slopes and intercepts

Since the slopes of the two lines (\(m_1 = m_2=-2\)) are equal, but the \(y\) - intercepts (\(b_1 = 4\) and \(b_2=\frac{3}{2}\)) are different, the two lines are parallel.
Parallel lines do not intersect, so the system of equations has no solution.

Answer:

C. The system is inconsistent.