Question

FFind an equation for the line that passes through the points (5,-2)(1,4)

Answer 1

y +2 = -3/2(x -5)

Explanation:

You want an equation for the line through the points (5, -2) and (1, 4).

Slope

One can start with the equation that says the slope is the same everywhere:

(y2 -y1)/(x2 -x1) = (y -y1)/(x -x1)

(4 -(-2))/(1 -5) = (y -(-2))/(x -5) . . . . substitute given points

-6/4 = (y +2)/(x -5) . . . . . . . . . . simplify

y +2 = -3/2(x -5) . . . . . . . . . point-slope equation (multiply by (x-5))

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Additional comment

This can be rearranged to standard form:

2y +4 = -3(x -5) . . . . . multiply by 2; next add 3x-4

3x +2y = 11 . . . . . . standard form equation of the line

The slope equation we started with cannot be used directly, as it is undefined at the point (5, -2).

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Answer 2

`y = -(3)/(2) x + (11)/(2)`

Explanation:

The equation of a line can be represented in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

Let's start by finding the slope of the line.

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

Notice how we are provided points. These points are always written in the format (x, y), which gives us the variables needed to find the slope.

`x_1=5\\x_2=1\\y_1=-2\\y_2=4`

Plugging these values into the equation, we get:

`m = (4-(-2))/(1-5)=(6)/(-4)=-(3)/(2)`

Now, we need to find the y-intercept. We can solve for the variable "b" by using any of the two points. Earlier, we realized that each point has an x and y value. This means we can use any of the two points given to solve for the y-intercept.

`y = -(3)/(2)x+b\\\\4=-(3)/(2)(1) + b\\\\4+(3)/(2)=b \Rightarrow b= (11)/(2)`

Therefore our equation is:

`y = -(3)/(2) x + (11)/(2)`