Question

Hello i have some questions i do not understand.

1. To the nearest tenth, what is the distance in units between the points (-2, 5) and (4,1)?

2. Which of the following shows a direct proportional relationship?
y =4x
y = 1/2x - 1
y = 8/x
y = x + 2/5

3.
What are the x- and y-intercepts of 6x - 4y = -36?

Answer 1

1) This is the equation for distance between two points:
`d= \sqrt{( {x _(1)} - x_(2))^(2) + ( {y _(1)} - y_(2))^(2)}`

Don't make it fool you, this is just a fancy translation of
`{a}^(2) + {b}^(2) = {c}^(2)` or the Pythogarean theorem.

The first parentheses or the x ones represents the difference between their location horizontally. Which is also a "leg".

The second or the y ones stands for the vertical difference.

Distance is the hypotenuse of those two values.

When we plug in the givens into the equation:
`d = \sqrt{ {(( - 2) - 4)}^(2) + {(5 - 1)}^(2) }`

Simplify:

`d = \sqrt{ { (- 6)}^(2) + {4}^(2) }`

`d = √(36 + 16)`

`d = √(52)`

2) The template for a "directly proportional" equation is:
`y = kx` where k is the constant which causes the "proportional" change.

The only one we see in the choices which is in this format is A.

3) To find x or y-intercepts, we need to plug in 0 for the other value.

For example, x-intercept of this equation is:
`6x - 4 * 0 = - 36`

Simplify:

`6x - 0 = - 36`

`6x = - 36`

`(6x)/(6) = ( - 36)/(6)`

`x = - 6`

So, the x-intercept is (-6, 0).

For y-intercept, the equation becomes:
`6 * 0 - 4y = - 36`

The zero cancels the six:
`- 4y = - 36`

When we divide both sides by -4:
`y = 9`

The y-intercept is (0, 9).