Question

How do you write the equation in point slope form given (3,5) and (8,15)?

Answer 1

`y=2x-1`

Explanation:

First, we have to solve for the slope using this equation:

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

This formula makes sense because slope is rise over run. Rise being the `y` values and run being the `x` values.

Now you choose which point is point 2 (includes `y_2` and `x_2`) and which point is point 1 (includes `y_1` and `x_1`)

Point 2: `(3,5)->y_2 = 5` and `x_2 =3`
Point 1: `(8,15)->y_1 = 15` and `x_1 = 8`

Plug into the formula and solve for the slope:

`m = (5-15)/(3-8)`

`m = (-10)/(-5)`

`m = (10)/(5)`

`m=2`

Now we know that the slope, `m=2`

Next, we have to use point-slope formula to get the equation:

`y-y_1=m(x-x_1)`

You can plug in either point for the `x_1` and `y_1` values. Let's use the point `(3,5)`.

`y-5=m(x-3)`

Now plug in the slope

`y-5=2(x-3)`

Distribute the `2`

`y-5=2x-6`

Add `5` to both sides

`y-5+5=2x-6+5`

`y=2x-1`

Final Answer:

`y=2x-1`

Answer 2

`color(indigo)(y - 5 = 2*(x - 3) " is the point - slope form of equation"`

Explanation:

If two points on a line are known, we can use the following formula to write the equation :

`(y - y_1) / (y_2 - y_1) = (x - x_1) / (x_2 - x_1)`

`"Given : (x_1, y_1) = (3,5), (x_2,y_2) = (8,15)`

Hence, equation of the line is

`(y - 5) / (15-5) = (x - 3) / (8 - 3)`

`(y-5) / cancel(10)^color(red)(2) = (x - 3) / cancel 5`

`color(purple)(y - 5 = 2*(x - 3)`

Standard form of Point-Slope equation is

`y - y_1 = m * (x - x_1)`

Answer 3

Equation of the line in Slope-Intercept Form: `color(green)(y=2x-1`

Explanation:

If the slope `(m)` of a line and the coordinate `(x_1, y_1)` of one end-point of the line is known, we can write the equation of the line in Point-Slope Form:

`color(red)(y-y_1=m(x-x_1)`

To find the slope if we are given two end-points of a line, we use the formula:

`color(blue)(Slope(m)=(y_2-y_1)/(x_2-x_1)`

We are given the points: `(3,5) and (8,15)`

Note that `x_1 = 3; y_1=5; x_2=8 and y_2=15`

`Slope(m)=(15-5)/(8-3)`

`rArr m=10/5`

`rArr Slope(m)=2`

Next, consider the equation of the point-slope form.

Consider the point `(3,5)`

We also found that `Slope(m)=2`

`x_1=3 and y_1=5`

`y-y_1=m(x-x_1)`

`rArr y-5=2(x-3)`

`rArr y-5=2x-6`

Add `5` to both sides of the equation.

`rArr y-5+5=2x-6+5`

`rArr y-cancel 5+cancel 5=2x-6+5`

`rArr color(green)(y=2x-1`

This is the required equation in the Point-Slope Form.

We can also graph the line and find the intercepts.

Substitute `y=0` to obtain the x-intercept.

`2x-1=0`

Add `1` to both sides.

`rArr 2x-1+1=0+1`

`rArr 2x-cancel 1+cancel 1=0+1`

`rArr 2x = 1`

Divide both sides by `2`

`(2x)/2=1/2`

`(cancel 2x)/cancel 2=1/2`

`x=1/2`

`x=0.5`

Hence `(0.5,0)` is the x-intercept.

Substitute `x=0` to obtain the y-intercept.

`y=2(0)-1`

`y=0-1`

`y=-1`

Hence `(0,-1)` is the y-intercept.