Question

Write an equation in point slope form that passes through the given points

1. (2,-2) (5,7)
2. (6,4) (2,1)

write an equation in standard form that passes through the given points
(0,3)(2,-3)
(1,-1) (4,2)

Answer 1

1.
`y+2 = 3(x-2)`

Point-slope form of the equation of a straight line is:

`y-y_0 = m(x-x_0)` (1)

The two points in this case are:

`(x_0,y_0)=(2,-2)\\(x_1,y_1)=(5,7)`

Slope of the line is given by:

`m=(y_1 -y_0)/(x_1 -x_0)=(7-(-2))/(5-2)=(9)/(3)=3`

Substituting into eq.(1), we find:

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

2.
`y-4 = (3)/(4)(x-6)`

Point-slope form of the equation of a straight line is:

`y-y_0 = m(x-x_0)` (1)

The two points in this case are:

`(x_0,y_0)=(6,4)\\(x_1,y_1)=(2,1)`

Slope of the line is given by:

`m=(y_1 -y_0)/(x_1 -x_0)=(1-4)/(2-6)=(3)/(4)`

Substituting into eq.(1), we find:

`y-4 = (3)/(4)(x-6)`

3.
`y+3x=3`

Standard form of the equation of a straight line is:

`ax+bx=c`

with a, b, c integer numbers.

Let's start by finding the point slope form first.

Point-slope form of the equation of a straight line is:

`y-y_0 = m(x-x_0)` (1)

The two points in this case are:

`(x_0,y_0)=(0,3)\\(x_1,y_1)=(2,-3)`

Slope of the line is given by:

`m=(y_1 -y_0)/(x_1 -x_0)=(-3-3)/(2-0)=-(6)/(2)=-3`

Substituting into eq.(1), we find:

`y-3 = -3(x-0)`

Now we can re-arrange the equation to re-write it in standard form:

`y-3 = -3(x-0)\\y-3 = -3x\\y-3+3x=0\\y+3x=3`

4.
`y-x=-2`

Standard form of the equation of a straight line is:

`ax+bx=c`

with a, b, c integer numbers.

Let's start by finding the point slope form first.

Point-slope form of the equation of a straight line is:

`y-y_0 = m(x-x_0)` (1)

The two points in this case are:

`(x_0,y_0)=(1,-2)\\(x_1,y_1)=(4,2)`

Slope of the line is given by:

`m=(y_1 -y_0)/(x_1 -x_0)=(2-(-1))/(4-1)=(3)/(3)=1`

Substituting into eq.(1), we find:

`y+1 = x-1`

Now we can re-arrange the equation to re-write it in standard form:

`y+1=x-1\\y+1-x=-1\\y-x=-2`