Question

Write the equation of a line that passes through the points (0, -2) and (4, -5). Answer MUST be in standard form: Ax + By = C.

Answer 1

Steps:

• Slope-Intercept Form: y = mx + b (m = slope, b = y-intercept)
• Standard Form: Ax + By = C (A, B, & C are integers and A must be non-negative)
• Slope Formula:
  `(y_2-y_1)/(x_2-x_1)` where (x₁,y₁) and (x₂,y₂) are coordinates.

So for this, I will be putting the equation into slope-intercept form then rearranging it into standard form. Firstly, we need to solve for the slope. To do this, take the 2 coordinates given to us and solve as such:

`(-5-(-2))/(4-0)=-(3)/(4)`

Now, one of the coordinates given to us is (0,-2), which is the y-intercept. With all this info, our slope-intercept equation is
`y=-(3)/(4)x-2` . From here we can solve for the standard form.

Firstly, add 3/4x onto both sides of the equation:

`(3)/(4)x+y=-2`

Now, it may appear that we are finished. However, 3/4 is not an integer (Integers are whole numbers). To make it an integer, we need to multiply both sides by 4:

`3x+4y=-8`

Answer:

In short, the standard form of this equation is 3x + 4y = -8.

Answer 2

The slope-intercept form:

`y=mx+b`

m - slope

b - y-intercept → (0, b).

We have the points (4, -5) and (0, -2) → y-intercept → b = -2.

The formula of a slope:

`m=(y_2-y_1)/(x_2-x_1)`

Substitute:

`m=(-2-(-5))/(0-4)=(-2+5)/(-4)=(3)/(-4)=-(3)/(4)`

Therefore we have the equation of a line

`y=-(3)/(4)x-2`

Convert to the standard form:
`Ax+By=C`

`y=-(3)/(4)x-2\qquad\text{multiply both sides by 4}\\\\4y=-3x-8\qquad\text{add 3x to both sides}\\\\\boxed{3x+4y=-8}`