Question

How do you write an equation in slope intercept form for the line through the given points (7,5 ); (-1, 1/5)?

Answer 1

`y = color(red)(3/5)x + color(blue)(4/5)`

Explanation:

First, we need to determine the slope. 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)(1/5) - color(blue)(5))/(color(red)(-1) - color(blue)(7)) = (color(red)(1/5) - (5/5 xx color(blue)(5)))/(color(red)(-1) - color(blue)(7)) = (color(red)(1/5) - 25/5)/(color(red)(-1) - color(blue)(7))`

`m = (-24/5)/-8 = 24/40 = (8 xx 3)/(8 xx 5) = (color(red)(cancel(color(black)(8))) xx 3)/(color(red)(cancel(color(black)(8))) xx 5) = 3/5`

Now we can use the point-slope formula to write an equation for the line. 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 from the problem gives:

`(y - color(red)(5)) = color(blue)(3/5)(x - color(red)(7))`

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. Solving the equation we found for `y` gives:

`y - color(red)(5) = (color(blue)(3/5) xx x) - (color(blue)(3/5) xx color(red)(7))`

`y - color(red)(5) = 3/5x - 21/5`

`y - color(red)(5) + 5 = 3/5x - 21/5 + 5`

`y - 0 = 3/5x - 21/5 + (5 xx 5/5)`

`y = 3/5x - 21/5 + 25/5`

`y = color(red)(3/5)x + color(blue)(4/5)`

Answer 2

`y=3/5x+4/5`

Explanation:

The equation of a line in `color(blue)"slope-intercept form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=mx+b)color(white)(2/2)|)))`
where m represents the slope and b, the y-intercept.

To calculate m, use the `color(blue)"gradient formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 coordinate points are `(7,5)" and (-1,1/5)`

let `(x_1,y_1)=(-1,1/5)" and " (x_2,y_2)=(7,5)`

`rArrm=(5-1/5)/(7+1)=(24/5)/8=3/5`

We can write the partial equation as `y=3/5x+b`

To find b, substitute either of the 2 given points into the partial equation and solve for b.

`"Using " (7,5)" that is " x=7,y=5`

`rArr5=(3/5xx7)+b`

`rArrb=5-21/5=25/25-21/25=4/5`

`rArry=3/5x+4/5" is equation in slope-intercept form"`