Answers and team assignments

  1.   What is the slope of 3x-4y=6 ?
      Let’s put it in the slope-intercept form (y=mx+b). That means we need to get y all by itself on the left side of the equation.

      Take away 3x from both sides: -4y = -3x + 6

Now divide everything by -4:     y = ¾ x - 6/4 or y = ¾ x – 3/2.

So the slope is ¾ .

  2.   What point is not on the line that the others are on?

(2,5), (3,8), (4,11), (6,17), (8,20)

There are several ways we can solve this. If you like to draw graphs, do that.

If you understand “slope = rise over the run” that would be easy.

Or if you like charts and patterns we could do that.

Lets do “rise over run.” Rise is how far up, run is how far to the right to the next point.

From (2,5) to (3,8) Rise is 8-5=3. Run is 3-2=1. Slope = 3/1.

From (3,8) to (4,11) Rise is 11-8=3. Run is 4-3=1. Slope = 3/1.

From (4,11) to (6,17) Rise is 17-11=6. Run is 6-4=2. Slope = 6/2 = 3/1.

From (6,17) to (8,20) Rise is 20-17=3. Run is 8-6=2. Slope = 3/2. Aha!! (8,20) is not on the same line!

33.    What is the equation of the line parallel to y = ¾ x – 2 that passes through the point (8,1)?

Parallel lines have the same slope, so our equation will be y = ¾ x + b. All we have to do is figure out b and we have an equation. But it passes through (8,1) which means our equation is true when x=8 and y=1.

So 1 = ¾ (8) + b.

Or 1 = 6 + b. therefore b must be -5. So the equation we were looking for is:

y = ¾ x – 5.

Let’s re-read the question. Is our equation’s line parallel to the given line? You bet – they both have a slope of ¾ . Does it pass through (8,1). Let’s double-check. 1 = ¾ (8) – 5. ¾ of 8 is 6 so does 1=6-5? Yes!

44. What is the equation of thee line perpendicular to y =  ¾ x – 2 that passes through the point (9,1)?
      This is similar to the previous problem but this time the line is perpendicular instead of parallel. To find the slope of the perpendicular, you take the negative reciprocal. So the slope of our line will be -4/3. The equation will be y = -4/3 x + b. We know when x=9, y=1 so to figure out b, we plug those values into our equation:

1 = -4/3 (9) + b

1 = -12 + b

13 = b so our equation is y = -4/3 x + 13

5.5.   Olivia sees a model of a pyramid that has a square base. The base of the pyramid was 6 inches long and the area of each one of the four sides was 12 sq. in. She would like to make her model larger. Her base will be a foot long.

a) If she builds it proportionally, what will be the total area of the 4 sides of her completed pyramid?

Doubling the length will quadruple the area. The area of a side was 12 sq.in. The new area of a side will be 4x12=48. The area of all 4 sides will be 4x48=192 sq. in.

b) How long is each of the 4 edges leading to the top of her pyramid?

Her pyramid has a base of 12” and area of 48 sq. in. Since the formula for the area of a triangle is A = ½ bh,

48 = ½ (12) h 

48 = 6 h

8 = height                  By dividing the triangle in two from top to bottom, we get a right triangle with base 6 and height 8. What is the edge? Use the Pythagorean theorem a²+b²=c²: 6² + 8²  = h^2.  Hence 36+64 = h² or 100 = h² and h the hypotenuse, which is the edge of the pyramid, = 10 in.

  

FRIENDLY HILLS
The only one above zero today was Sophia with 2 points.
These problems required algebra.

Oscar Halverson	50.0
Cyrus Martin Ri	36.0
Nick Wendt	30.0
Sophia Schomer	25.0
Nina Kessler	18.0
Miles Dunn	17.0
Dain Dolan	15.0
Will Gannon	15.0
Duy Houng	13.0
Amario Sisakda	12.0
Lucas Lindgren	11.0
Lily Pince	11.0
Ellyanna Lee	10.0
Charles Cheesebrough	10.0
Melissa Irwin	8.0
Claire Newmark	7.0
Jackson Cercioglu	6.0
Stella Warwick	6.0
Victoria Dzurilla	2.0

HERITAGE
The only ones above zero today were Anthony with 12 and Ben with 2.

Anthony Rocke	69.0
Kathryn Lewis	31.0
Ben Nickson	30.0
Steve Nickson	24.0
Aidan Mallberg	13.0
Emma Lawrence	13.0
Colten Bartlette	5.0
Dia Balderramos	6.0
Maraya Lucio	4.0
Alycia Gonzales Lewis	4.0
Ruby Lipschultz	6.0
Sophie Todaro	2.0