Question

1. which function has the smallest slope?

$y = -\frac{3}{2}x - 4$
$y = -\frac{2}{3}x + 2$
$3x + y = -9$
$9x + 2y = 5$

Explanation:

Step1: Recall slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. For equations not in this form, we solve for \( y \) to find \( m \).

Step2: Find slope of \( y = -\frac{3}{2}x - 4 \)

Here, the slope \( m_1 = -\frac{3}{2}=-1.5 \).

Step3: Find slope of \( y = -\frac{2}{3}x + 2 \)

Here, the slope \( m_2 = -\frac{2}{3}\approx - 0.67 \).

Step4: Find slope of \( 3x + y=-9 \)

Solve for \( y \): \( y=-3x - 9 \). So, slope \( m_3=-3 \).

Step5: Find slope of \( 9x + 2y = 5 \)

Solve for \( y \): \( 2y=-9x + 5\Rightarrow y =-\frac{9}{2}x+\frac{5}{2} \). So, slope \( m_4 =-\frac{9}{2}=-4.5 \).

Step6: Compare the slopes

We have \( m_1=-1.5 \), \( m_2\approx - 0.67 \), \( m_3 = - 3 \), \( m_4=-4.5 \). Arranging them in ascending order: \( - 4.5<-3<-1.5<-0.67 \). So the smallest slope is \( - 4.5 \) (i.e., \( -\frac{9}{2} \)) which corresponds to the line \( 9x + 2y = 5 \).

Answer:

The function \( 9x + 2y = 5 \) has the smallest slope.