an equation in slope-intercept form of the line that passes through (−2, 5) and (−4, −5)

Answer 1

The standard form for a linear equation is:

y = mx + b

Where:

1. m = the slope
2. b = the y-intercept (the point where the line intercepts the y-axis)
3. x = the independent variable
4. y = the dependent variable

First, one must find the slope. To do this, one must first remember that the slope is defined as:

"rise over run"

Therefore, one must find the differences between the x coordinates and y coordinates of the two points as follows:

1. P₁ = ( x₁, y₁ ) = ( -2, 5 )
2. P₂ = ( x₂, y₂ ) = ( -4, -5 )
3. It does not matter from which point one subtracts the other, the slope will come-out to be the same:
4. Method A:
5. Δx = x₁ - x₂ = -2 - ( -4 ) = -2 + 4 = 2
6. Δy = y₁ - y₂ = 5 - ( -5 ) = 5 + 5 = 10
7. m = Δy / Δx = 10 / 2 = 5
8. Method B:
9. Δx = x₂ - x₁ = -4 - ( -2 ) = -4 + 2 = -2
10. Δy = y₂ - y₁ = -5 - 5 = - 10
11. m = Δy / Δx = -10 / -2 = 5

Now that the slope has been found, the equation becomes:

1. y = mx + b
2. y = 5x + b; using the value of the slope found in the previous step

The next thing to do is to find the value of b, which is the y-intercept. To do this, one can use either one of the two points provided in the problem:

1. Method A (Using P₁)
2. P₁ = ( x₁, y₁ ) = ( -2, 5 )
3. x = -2
4. y = 5
5. y = 5x + b
6. 5 = (5) ( -2 ) + b
7. -10 + b = 5
8. b = 15
9. Method B (Using P₂):
10. P₂ = ( x₂, y₂ ) = ( -4, -5 )
11. x = -4
12. y = -5
13. y = 5x + b
14. -5 = (5) ( -4 ) + b
15. -20 + b = -5
16. b = 15

Using both P₁ and P₂, it was found that the y-intercept of the line is b = 15. Using this last value, and substituting it for b in the equation, this equation now becomes:

1. y = mx + b
2. y = 5x + b
3. y = 5x + 15

FINAL ANSWER:

y = 5x + 15

Answer 2

First you need to find the slope (m) using the two points given:

m = (y2 - y1)/(x2 - x1) = (-5 - 5)/(-4 -(-2)) = (-10)/(-2) = 5

Then use this slopes and any one of the two given points to write the equation in point-slope form:

(y - y1) = m(x - x1)

y - 5 = 5(x - (-2))

y - 5 = 5(x + 2)

Then isolate the y to write the equation in slope-intercept form:

y - 5 = 5x + 10 add 5 to both sides

y = 5x + 15

You can double check that you didn't make a mistake by plugging in each point in the final equation. If it works out then you got it right.

Using (-4, -5)

y = 5x + 15

-5 = 5(-4) + 15

-5 = -20 + 15

-5 = -5 this checks out

Using (-2, 5)

y = 5x + 15

5 = 5(-2) + 15

5 = -10 + 15

5 = 5 this checks out