~ Find the distance from the point (-2, 1) to the line x-y -2=0

The shortest distance is on the line through the point (-2,1) that is perpendicular to the lilne “x-y-2=0).

First, rewrite the equation of the given line in the slope-intercept form: y = mx + b … in this case you get y = x – 2 (with m =1 and b = -2)

Given a line of slope “m”, the lines that are perpendicular to it have a slope of “-1/m” so your perpendicular line is y=-1*x +b or … y=-x +b. Plug the given point (-2,1) into this equation to figure out your “b” value.

1 = -(-2) +b ===> 1 = 2 +b ===> b = -1

The line perpendicular to the original line (y = x – 2) through the given point (-2,1) is y= -x -1

You now have two equations for two lines. The point of intersection is found by solving the two equations in two unknowns.

#1) y= x – 2

#2) y= -x – 1

[add #2 to #1 to get) 2y = -3 ====> y = -3/2 ===> y = -1.5

[plug -1.5 into either #1 or #2 to get x] -1.5 = x -2 ===> 0.5 = x

The intersecting point is (0.5, -1.5)

Now, calculate the distance between the two points: (-2,1) and (0.5, -1.5). The general formula is distance = sqrt( (x1-x2)^2 + (y1-y2)^2) ) so …

distance = sqrt ( (-2 -.5)^2 + (1 – -1.5)^2) )

distance = sqrt ( (-2.5)^2 + (2.5)^2) )

distance = sqrt ( 6.25 + 6.25 ) = sqrt (12.5)

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~ A house was purchased 10 years ago for $60,000. This year it was appraised at $85,000. ….

Value 10 years ago: V(0) = 60,000

Value now: V(10) = 85,000

You are given that V(t) = mt+b (looks like the slope-intercept equation of a line)

A.) Find an equation for V(t).

– For 10 years ago, V(0) = m*0 + b ===> 60,000 = 0 + b ===> b = 60,000

– Today, V(10) = m*10 + b ===> 85,000 = 10*m + 60000 ===> 25,000 = 10*m ===> 2,500 = m

So, V(t) = 2,500*t + 60,000

B.) When was the house worth $71,250?

Since 71,250 = 2,500*t + 60,000, solve for t.

71,250 – 60,000 = 2,500t

11,250 = 2500t

4.5 = t

The house was worth 71,250 4.5 years after it was purchased (10 years ago) so that would be 5.5 years ago.

When you want to do things with parentheses like this, you have to do what’s called distribution. Let’s start with the first one – since the X and the 4 are both within parentheses and the 11 is being multiplied by the whole thing, you have to ‘share’ the 11 with everything that’s inside. That is what distributing means. To get the answer, you multiply everything in the parentheses by the 11. Writing it out would look like this: 11(x – 4) (11*x – 11*4) (11x – 44) It’s a general rule that whenever you are dealing with parentheses you have to distribute. Try the other problems yourself so you get some practice before the test.