Question

1. given the points (4,-3) and (2,4):

a. write the equation of the line that contains both points
b. does the point (8, -17) lie on the same line as the points given above?

Explanation:

Step1: Calculate the slope

Let $(x_1,y_1)=(4,-3)$ and $(x_2,y_2)=(2,4)$.
Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{4-(-3)}{2-4}=\frac{7}{-2}=-\frac{7}{2}$

Step2: Find y-intercept $b$

Use point-slope form $y-y_1=m(x-x_1)$, substitute $(4,-3)$ and $m=-\frac{7}{2}$:
$y-(-3)=-\frac{7}{2}(x-4)$
Simplify to slope-intercept form $y=mx+b$:
$y+3=-\frac{7}{2}x+14$
$y=-\frac{7}{2}x+11$

Step3: Verify point (8,-17)

Substitute $x=8$ into the line equation:
$y=-\frac{7}{2}(8)+11=-28+11=-17$

Answer:

a. The equation of the line is $y=-\frac{7}{2}x+11$ (or in standard form: $7x+2y=22$)
b. Yes, the point $(8, -17)$ lies on the same line.