Question

which equation represents the line that passes through (-8, 11) and (4, \frac{7}{2})?
\\( y = -\frac{15}{2}x - 49 \\)
\\( y = -\frac{5}{8}x + 16 \\)
\\( y = -\frac{5}{8}x + 6 \\)
\\( y = -\frac{15}{2}x + 71 \\)

Explanation:

Step1: Calculate the slope (m)

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1)=(-8, 11)\) and \((x_2, y_2)=(4, \frac{7}{2})\). Then \(m=\frac{\frac{7}{2}-11}{4 - (-8)}=\frac{\frac{7 - 22}{2}}{12}=\frac{-\frac{15}{2}}{12}=-\frac{15}{24}=-\frac{5}{8}\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Using the point \((-8, 11)\) and \(m =-\frac{5}{8}\), we have \(y - 11=-\frac{5}{8}(x + 8)\).
Expand the right - hand side: \(y-11=-\frac{5}{8}x-5\).
Add 11 to both sides: \(y=-\frac{5}{8}x-5 + 11\), so \(y=-\frac{5}{8}x + 6\). We can also check by plugging in the other point \((4,\frac{7}{2})\) into the equation \(y =-\frac{5}{8}x+6\). When \(x = 4\), \(y=-\frac{5}{8}\times4 + 6=- \frac{5}{2}+6=\frac{-5 + 12}{2}=\frac{7}{2}\), which matches the y - coordinate of the second point.

Answer:

\(y =-\frac{5}{8}x + 6\) (the option \(y =-\frac{5}{8}x + 6\))