Question

Question #0443e

Answer 1

`1a)   x - y = 2`
`1b) color(white)(.)x + y =` - `2`

`2a)  x - y =` - `3`
`2b)   x + y = 1`

Explanation:

1a) Parallel to `y = x - 3` through `(0,`-`2)`

Parallel lines have the same slope.
The point P is `b`, the `y` intercept.

So this line, parallel to the given line, is
`y = x - 2`

Standard form
`x - y = 2`
~ ~ ~ ~ ~ ~ ~

1b) Perpendicular to `y = x - 3` through `(0,`-`2)`

The point P is `b`, the `y` intercept.
The slopes of lines perpendicular to each other are the negative inverses of each other.
So the slope of the unknown line is -`1.`

This line, perpendicular to the given line, is
`y = - x - 2`

Standard form
`x + y = - 2`
~ ~ ~ ~ ~ ~ ~ ~ ~

2a) Parallel to `y = x - 3` through (-`1,2)`

Because the lines are parallel, both slopes are 1.

To find `b`, sub in the values for `x` and `y` and solve for `b`

`y =  m    x  + b`
`2 = (1)` `(`-1) `+ b`
`2 = color(white)(.)` -`1      + b`
`3 = b`
`y = x + 3`

Standard form
`x - y =` - `3`
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

2b) Perpendicular to `y = x - 3` through (-`1,2)`
The perpendicular slope will be -`1`
The new `y` intercept will be
`y =   m (x) + b`
`2 =` *-1 (*-`1)` `+ b`
`2 = 1 + b`
`1 = b`
`y =` -`x + 1`

Standard form
`x + y = 1`