Question

write the equation of the line that passes through the points (-4, 5) and (2, -4). put your answer in fully simplified point - slope form, unless it is a vertical or horizontal line.

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1)=(-4, 5) \) and \( (x_2, y_2)=(2, -4) \). So, \( m=\frac{-4 - 5}{2 - (-4)}=\frac{-9}{6}=-\frac{3}{2} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use either of the two points. Let's use the point \( (-4, 5) \). Substituting \( m = -\frac{3}{2} \), \( x_1=-4 \), and \( y_1 = 5 \) into the point - slope formula, we get \( y - 5=-\frac{3}{2}(x - (-4)) \), which simplifies to \( y - 5=-\frac{3}{2}(x + 4) \). We can also use the point \( (2,-4) \). Substituting \( m = -\frac{3}{2} \), \( x_1 = 2 \), and \( y_1=-4 \) into the point - slope formula, we get \( y-(-4)=-\frac{3}{2}(x - 2) \), or \( y + 4=-\frac{3}{2}(x - 2) \). Both are correct, but let's check with the first point. Expanding \( y - 5=-\frac{3}{2}(x + 4) \), we have \( y-5 = -\frac{3}{2}x-6 \), and then \( y=-\frac{3}{2}x - 1 \). But in point - slope form, using the first point: \( y - 5=-\frac{3}{2}(x + 4) \) is a fully simplified point - slope form.

Answer:

\( y - 5=-\frac{3}{2}(x + 4) \) (or \( y + 4=-\frac{3}{2}(x - 2) \))