Cows and Chickens
A farmer counted the number of cows and chickens by counting heads and legs. If he counted 35 heads and 78 legs, how many cows and how many chickens did he have?
In the Introduction, you were given a problem to experiment with. If you were lucky you might find the answer right away, but as problems get more difficult, using trial and error could also be futile.
Let’s look more closely at this problem.
A farmer had 35 heads of cows and chickens on his farm. Barring any two-headed cows or chickens, this would tell us there are 35 animals total. That might be 1 and 34, 2 and 33, or 3 and 32 and so on.
We could also create an equation for this:
Let x = cows and
y = chickens.
Cows + Chickens = 35
so x + y = 35.
A good way to represent this would be to graph it. Plot the line, starting with the point (0, 35).
Answer:
The farmer also had 78 legs on the cows and chickens. Cows have four legs and chickens have two legs.
4x + 2y = 78
That might be 4(1 cow) and 2(37 chickens), 4(2 cows) and 2(35 chickens), 4(3 cows) and 2(33 chickens), and so on. Plot the line above on the same graph.
Answer:
All points on the blue line will make the first equation true and all points on the red line will make the second equation true, but the only answer that is a solution to the system is the point of intersection which is true for both equations. That point is (4,31) which represents 4 cows and 31 chickens.
4 + 31 = 35 heads and
4(4) + 2(31) = 78 legs.
View solution here.
Example #1
Troy and his friends meet for breakfast at a local snack bar before school each morning. One morning Troy bought three biscuits and two drinks for $9.00. Jarrod spent $14.50 on five biscuits and three drinks. Create a system of equations to represent the problem and find the cost of the biscuits and the drinks by graphing.
Let’s start by identifying the two variables. They are biscuits and drinks. Represent biscuits with the letter b and drinks with the letter d.
Let's start by looking at Troy’s breakfast and finding an equation for it.
_ b + _ d = $__
Answer: 3b + 2d = $ 9.00
Now, find an equation representing Jarrod's breakfast.
$_ = _ b + _ d
Answer: $14.50 = 5b + 3d
Troy: 3b + 2d = $9.00
Jarrod: 5b + 3d = 14.50
Let’s start by graphing the equation for Troy’s breakfast. Let b be the horizontal axis and d the vertical axis.
Substitute in 1, 2, and 3 for the value of b and find the value of d. Graph those points to create a line.
Answer:
Now, graph the equation for Jarrod’s breakfast. Let b be the horizontal axis and d the vertical axis.
Try using 0.50, 1.70, and 2.3 for the values of b.
Answer:
The solution is the intersection. Biscuits were $2.00 and drinks were $1.50.
View solution here.
