Question

rite the point - slope form of the equation of the line through the given points.
through: (4, 4) and (1, -1)

1. through: (-1, 1) and (0, 5)

Explanation:

For the first sub - question (through: (4, 4) and (1, - 1))

Step 1: Calculate the slope

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(4,4)$ and $(x_2,y_2)=(1, - 1)$. Then $m=\frac{-1 - 4}{1 - 4}=\frac{-5}{-3}=\frac{5}{3}$.

Step 2: Write the point - slope form

Using the point - slope formula $y - y_1=m(x - x_1)$ and the point $(4,4)$, we get $y - 4=\frac{5}{3}(x - 4)$. We could also use the point $(1,-1)$: $y+1=\frac{5}{3}(x - 1)$.

For the second sub - question (through: (- 1, 1) and (0, 5))

Step 1: Calculate the slope

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(-1,1)$ and $(x_2,y_2)=(0,5)$. Then $m=\frac{5 - 1}{0+1}=\frac{4}{1}=4$.

Step 2: Write the point - slope form

Using the point - slope formula $y - y_1=m(x - x_1)$ and the point $(-1,1)$, we get $y - 1=4(x + 1)$. We could also use the point $(0,5)$: $y - 5=4(x - 0)$ or $y - 5 = 4x$.

Answer:

For the line through (4, 4) and (1, - 1): $y - 4=\frac{5}{3}(x - 4)$ (or $y+1=\frac{5}{3}(x - 1)$)
For the line through (- 1, 1) and (0, 5): $y - 1=4(x + 1)$ (or $y - 5=4x$)