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Systems Problem of the Week
Transcript of Systems Problem of the Week
For What Values of K will the graphs of the following equations intersect in Quadrant 3?
The first way to solve the problem is to graph it out visually. This can give you a better view of what you are working with.
? < k < 1
Summery: A carpenter to 2 hardware stores to buy a certain number of pounds of nails. Store #1 is selling nails for 9 cents per pound, with this price, he would be short $7.15 if wanted to buy the nails. Store #2 sells nails for 6 cents, if he buys these nails, he would have $2.45 left. How many pounds of nails is he trying to buy?
Method of Solution:
To make everything more organized and clear, I created a table to represent my information.
Systems Problem of the Week
Line 1: y = kx + 14
Line 2: y = x + 28
What I know:
What Information I need:
- The intersection must be in quadrant 3
- x and y must be a negative number (because the values in quadrant 3 are always negative)
- x in line 2 must equal less than -28 (Line 2: y = x + 28)
- y has a greater value than x
-Since line 1 is the line with the variable, k, it will be the only line affected and that will change along with the different values of K
- What values can be imputed into K, for x and y to intersect in Quadrant 3
- What are the values of x and y
k cannot equal 1. This is because if it does, the lines will not even intersect in any quadrants, it will be parallel to each other and will never intersect.
So What's after that?
Now that we know that k will not work with 1 and over or under, what we need to know now is whether its over or under 1. We can do this by graphing the lines with k as 0.9 and 1.1, to see which will work so we know which direction to go to.
Substituting K with 0.9
If k has the value of 0.9, then the two lines will intersect in Quadrant 3.
Substituting k with 1.1:
Although k works for 0.9, just to make sure, lets make sure it will work for 1.1, as you can see, the lines intersects in the 1nd quadrant, which proves that k must be below 1 for the question to be true.
Now what we need to find is the 2nd boundary for what k cannot be.
0.5 < K < 1
K must be larger than .5 but smaller than 1 for the 2 lines to intersect in quadrant 3.
y = 9x - 7.15 and y = 6x + 2.45 ---> 9x - 7.15 = 6x + 2.45
9x - 7.15 = 6x + 2.45
9x = 6x + 9.6
3x = 9.6
x = 3.2
The carpenter is trying to buy 320 pounds of nails.
9(3.2) - 7.15 = 6(3.2) + 2.45
28.8 -7.15 = 19.2 + 2.45
21.65 = 21.65
Since x = 3.2 when the units are cents, they need to be multiplied by 100 to make the units dollars
3.2 * 100 = 320