Question

Complete the equation of the line through-6, 5) and(3, -3). Use exact numbers.​

Answer 1

Explanation:

(-6,5) (3,-3)

The equation of the line:

`\boxed {(x-x_1)/(x_2-x_1)=(y-y_1)/(y_2-y_1) }`

x₁=-6 x₂=3 y₁=5 y₂=-3

Hence,

`\displaystyle\\(x-(-6))/(3-(-6)) =(y-5)/(-3-5) \\\\(x+6)/(3+6)=(y-5)/(-8) \\\\(x+6)/(9) =(y-5)/(-8) \\\\`

Multiply both parts of the equation by -72:

`-8(x+6)=9(y-5)\\\\-8x-48=9y-45\\\\-8x-48+45=9y-45+45\\\\-8x-3=9y\\\\`

Divide both parts of the equation by 9:

`\displaystyle\\-(8)/(9) x-(1)/(3) =y\\\\Thus,\ y=-(8)/(9)x-(1)/(3)`

Answer 2

To find the equation of the line through the two given points, we can use the slope-intercept form of the equation of a line, which is $y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept (the point where the line crosses the $y$-axis).

We can find the slope of the line by using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the two given points. In this case, the slope is

$$m = \frac{(-3) - 5}{3 - (-6)} = \frac{-8}{9} = -\frac{4}{3}.$$

To find the $y$-intercept, we can plug the coordinates of one of the points and the slope into the equation of a line, so we have $y = -\frac{4}{3} x + b$. If we plug in the coordinates of the first point, we get

$$5 = -\frac{4}{3} (-6) + b \Rightarrow b = -\frac{4}{3} (-6) + 5 = \frac{28}{3}.$$

Therefore, the equation of the line through the two given points is

$$y = -\frac{4}{3} x + \frac{28}{3} = -\frac{4}{3} x + \frac{28}{3}.$$