Question

write an equation of the line that passes through the pair of points. use improper fractions in your equation, if necessary. (5, -3), (2, 5)

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((5, - 3)\) and \((2,5)\), we have \( x_1 = 5,y_1=-3,x_2 = 2,y_2 = 5 \). So \( m=\frac{5-(-3)}{2 - 5}=\frac{5 + 3}{-3}=\frac{8}{-3}=-\frac{8}{3} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((2,5)\) (we could also use \((5,-3)\)). Substitute \( m =-\frac{8}{3},x_1 = 2,y_1 = 5 \) into the point - slope form: \( y-5=-\frac{8}{3}(x - 2) \).

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y-5=-\frac{8}{3}x+\frac{16}{3} \). Then add 5 to both sides. Since \( 5=\frac{15}{3} \), we have \( y=-\frac{8}{3}x+\frac{16}{3}+\frac{15}{3} \). Combine the fractions: \( y=-\frac{8}{3}x+\frac{31}{3} \).

Answer:

\( y =-\frac{8}{3}x+\frac{31}{3} \)