Question

how many solutions does the system of equations below have?
x + 2y = -3
2x + 4y = 7
no solution
one solution
infinitely many solutions
submit

Explanation:

Step1: Analyze the slopes of the lines

First, rewrite the equations in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

For the equation \(x + 2y=-3\):
Subtract \(x\) from both sides: \(2y=-x - 3\)
Divide both sides by 2: \(y=-\frac{1}{2}x-\frac{3}{2}\). The slope \(m_1 =-\frac{1}{2}\)

For the equation \(2x + 4y = 7\):
Subtract \(2x\) from both sides: \(4y=-2x + 7\)
Divide both sides by 4: \(y =-\frac{2}{4}x+\frac{7}{4}=-\frac{1}{2}x+\frac{7}{4}\). The slope \(m_2=-\frac{1}{2}\)

Step2: Analyze the y - intercepts

The y - intercept of the first line \(b_1=-\frac{3}{2}\) and the y - intercept of the second line \(b_2 = \frac{7}{4}\)

Since the slopes of the two lines are equal (\(m_1=m_2 =-\frac{1}{2}\)) and the y - intercepts are different (\(b_1
eq b_2\)), the two lines are parallel and do not intersect.

Answer:

no solution