The equations for the line shown below are B. y + 5 = -3(x - 1) and D. y - 4 = -3(x + 2).
To determine which equations represent the line passing through the points (-2, 4) and (1, -5), we can use the point-slope form of a linear equation:
![\[y - y_1 = m(x - x_1)\]](https://img.qammunity.org/2022/formulas/mathematics/college/8ap9z4mrjt5mlk1d0sk7u7hzmfea5uswz0.png)
where
is a point on the line, and \(m\) is the slope.
Given the points (-2, 4) and (1, -5), we can find the slope:
![\[m = (y_2 - y_1)/(x_2 - x_1)\]](https://img.qammunity.org/2022/formulas/mathematics/college/i8dr567b7h7mt95b00ky28ahfhiq6z7hcg.png)
![\[m = (-5 - 4)/(1 - (-2)) = (-9)/(3) = -3\]](https://img.qammunity.org/2022/formulas/mathematics/college/hv8wvvtqo58z2elckqchwmq6dvfrejhov1.png)
Now, let's use the point-slope form with one of the points:
![\[y - 4 = -3(x + 2)\]](https://img.qammunity.org/2022/formulas/mathematics/college/cf88o7pxhu4tz1kocyyu7ej97fgvmeb7lz.png)
Now, let's compare this with the given options:
A. y = -3x - 3 - Not equivalent.
B. y + 5 = -3(x - 1) - Equivalent after simplification.
C. y = -3x - 2 - Not equivalent.
D. y - 4 = -3(x + 2) - Equivalent.
So, the correct choices are:
- B. y + 5 = -3(x - 1)
- D. y - 4 = -3(x + 2)