Question

a linear relationship is shown in the table.

x	0	2	6
y	-5	1	13

part a: determine the slope of the linear relationship. show all necessary work. (4 points)
part b: write the equation of the line in slope-intercept form. explain. (4 points)
part c: is the linear relationship also a proportional relationship? why or why not? (4 points)

Explanation:

Part A:

Step1: Identify coordinate pairs

Use points $(0, -5)$, $(2, 1)$, $(6, 13)$

Step2: Apply slope formula

Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$. Use $(0, -5)$ and $(2, 1)$:
$m = \frac{1 - (-5)}{2 - 0} = \frac{6}{2} = 3$
Verify with $(2, 1)$ and $(6, 13)$:
$m = \frac{13 - 1}{6 - 2} = \frac{12}{4} = 3$

Part B:

Step1: Find y-intercept

From $(0, -5)$, $b = -5$

Step2: Write slope-intercept form

Slope-intercept form is $y = mx + b$. Substitute $m=3$, $b=-5$:
$y = 3x - 5$

Part C:

Step1: Recall proportionality rule

Proportional lines pass through $(0,0)$ (y-intercept $b=0$)

Step2: Compare to our line

Our line has $b = -5
eq 0$, so it is not proportional.

Answer:

Part A: The slope is $3$
Part B: The equation is $y = 3x - 5$; the slope is 3, and the y-intercept (from $x=0$) is $-5$, which fits the slope-intercept form $y=mx+b$.
Part C: No, it is not proportional. A proportional linear relationship must pass through the origin $(0,0)$, but this line has a y-intercept of $-5$, so it does not pass through the origin.