Question

How do you write an equation in slope-intercept form for a line with points (-3, 1) and (-2, -5)?

Answer 1

See the entire solution process below:

Explanation:

First, we need to determine the slope of the line passing through these two points. The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(-5) - color(blue)(1))/(color(red)(-2) - color(blue)(-3)) = (color(red)(-5) - color(blue)(1))/(color(red)(-2) + color(blue)(3)) = -6/1 = -6`

Next, we can write an equation in point-slope form. The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through. Substituting the slope we calculated and the first point gives:

`(y - color(red)(1)) = color(blue)(-6)(x - color(red)(-3))`

`(y - color(red)(1)) = color(blue)(-6)(x + color(red)(3))`

Now, solve for `y` to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: `y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b)` is the y-intercept value.

`y - color(red)(1) = (color(blue)(-6) xx x) + (color(blue)(-6) xx color(red)(3))`

`y - color(red)(1) = -6x - 18`

`y - color(red)(1) + 1 = -6x - 18 + 1`

`y - 0 = -6x - 17`

`y = color(red)(-6)x - color(blue)(17)`

Answer 2

`y = -6*x - 17`

Explanation:

y = mx + b Calculate the slope, m, from the given point values, solve for b by using one of the point values, and check your solution using the other point values.

A line can be thought of as the ratio of the change between horizontal (x) and vertical (y) positions. Thus, for any two points defined by Cartesian (planar) coordinates such as those given in this problem, you simply set up the two changes (differences) and then make the ratio to obtain the slope, m.

Vertical difference “y” `= y_2 – y_1 = -5 – 1 = -6`
Horizontal difference “x”`= x_2 – x_1 = -2 – -3 = 1`
Ratio = “rise over run”, or vertical over horizontal`= ((-6)/1) = -6` for the slope, m.

A line has the general form of y = mx + b, or vertical position is the product of the slope and horizontal position, x, plus the point where the line crosses (intercepts) the x-axis (the line where z is always zero.) So, once you have calculated the slope you can put any of the two points known into the equation, leaving us with only the intercept 'b' unknown.

`1 = (-6)*(-3) + b ; 1 = 18 + b ; 1 – 18 = b ; -17 = b`

Thus the final equation is `y = -6*x - 17`
We then check this by substituting the other known point into the equation:
`-5 = (-6)*(-2) - 17 ; -5 = 12 - 17 ; -5 = -5` CORRECT!