Jamie wants to know which hill slope his billy cart will travel fastest down and which is longest. Hill 1 has the equation -2x - 4 = -2y + 6x and Hill 2 has the equation y = 3x + 2.5. The assistant calculates that Hill 2 has the steeper slope and Jamie will travel fastest on it. They also calculate that the length of Hill 1 is 2.06km and Hill 2 is 7.55km, so Hill 2 is the longest slope.

1. Challenging Algebra
Question

2. Young Jamie decided to test his new non-motorised billy kart on two different
straight hill slopes. He only had the mathematical equations for the slopes, but
wanted to know how far he would travel on his billy cart and which hill he would
go down the fastest on.
Hill 1:
Hill 2:
-2x - 4 = -2y + 6x
x=-
y
+ 4 x + 2.5
3
Can you help Billy with his Mathematics equations. Both hills have perfectly
straight and even roads with no bumps. He needs you to:
Find out which hill will generate the fastest speeds on his billy kart.
Which straight hill slope is the longest stretch of road if he starts at the
x-intercept and the finish line is at the y-intercept of those lines? Assume that
both lengths of roads are using the same unit of measurement (kms).

3. Slope 1
-2x - 4 = -2y + 6x
-2x - 6x - 4 = -2y
-8x - 4 = -2y
-8x - 4
y=
-2
y = 4x + 2
Slope 2
y
x = - + 4x + 2.5
3
y
= 4x - x + 2.5
3
y
= 3x + 2.5
3
y = 3(3x + 2.5)
y = 9x + 7.5
Slope 2 has the steeper gradient, so Jamie will travel the fastest speeds on this
Slope.

4. Part 2 of the Question
Which straight hill slope is the longest stretch of road if he starts at
the y-intercept and the finish line is at the x-intercept of those lines?
y = 4x + 2
y-intercept …x = 0
Y intercept = 2
X intercept ..y = 0
y =- 1/2
2
1
- + 22 = c2
2
.25 + 4 = c 2
c 2 = 4.25
c = 4.25
c = 2.06
The length of this slop 1 is 2.06 kms

5. Slope 2
y = 9x + 7.5
When x=0
Y=7.5
When y=0
7.5
9
-0.833
0.8332 + 7.52 = c 2
0.694 + 56.25 = c 2
56.944 = c 2
c = 56.944
c = 7.55
(Rounded to 2 Decimal Places)
The length of slope 2 = 7.55kms

6. Solution
The length of this slop 1 is 2.06kms
The length of slope 2 = 7.5 kms
Therefore, the longest slope is going to
be slope 2.