To find the equation of the line that goes through the points \((-4, 5)\) and \((8, 14)\), we first need to calculate the slope of the line. The slope (m) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in the coordinates of the two points: \((x_1, y_1) = (-4, 5)\) and \((x_2, y_2) = (8, 14)\), we have:
\[ m = \frac{14 - 5}{8 - (-4)} = \frac{9}{12} = \frac{3}{4} \]
Now that we have the slope \(m = \frac{3}{4}\), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using one of our points, say \((-4, 5)\):
\[ y - 5 = \frac{3}{4}(x - (-4)) \] \[ y - 5 = \frac{3}{4}(x + 4) \]
Next, we simplify this equation:
\[ y - 5 = \frac{3}{4}x + 3 \] \[ y = \frac{3}{4}x + 8 \]
Now, let's compare this with the options provided:
- \(y = \frac{3}{4}x + 5\)
- \(y = \frac{4}{3}x + 5\)
- \(y = \frac{3}{4}x + 8\)
- \(y = \frac{4}{3}x + 8\)
The correct equation for the line that goes through the points \((-4, 5)\) and \((8, 14)\) is:
\[ \boxed{y = \frac{3}{4}x + 8} \]