write a system of two equation and use them to solve the following problem: two farmers were discussing how many chickens and cows they would purchase at market. the first farmer says to the second, I have just purchased 30 new heads and 100 feet. how many chicken and how many cows did the farmer purchase?
X new chickens
Y new cows.
X + Y = 30
2X + 4Y = 100
2X + 2Y = 60
2Y = 40
Y = 20
X = 30 - Y = 10
Perhaps the third farmer was too young to have understood at the time that the ninths were a ridiculous suggestion? When he eventually grew up and learned math, he probably demanded that the third farmer give him the third cow but was denied and eventually tried to steal the one he felt was his from the second farmer, but was caught and was punished severely. After being banished he nearly lost his way...until he met a green alien named Toad who taught him the ways of Frog. He learned that kindness, compassion, courage, equanimity, humility, gratitude, wisdom, loyalty, perseverance, tolerance, self-control, Self awareness, Emptiness and generosity were the best way to practice farming. Years later, he gifted the second farmer 18 cows and asked him to share them with his family by any ratio that that he felt was fair...halves, thirds, quarters, etc. whatever you think is reasonable. The third farmer responded by giving a cow to every person in his community who were struggling and said, “These people are my family, they are very grateful for your generosity. I am very sorry that I overreacted and pushed you away for all those years. You won’t believe this but I met an alien called Frog who taught me the ways of Toad, where I learned that everyone is an extension of ourselves...Self. When we can help people in need we can feel alive while helping countless beings! I am very sorry for what I did to you so many years ago. I often wish I could take it back. Is there anything I can do to make it up to you?”
The third farmer replied, “You don’t have to make it up to me. We all make mistakes. A genuine apology goes a long way, and I must say that I am sorry I was not able to restrain myself at the time. I have also happened upon a little green alien named Toad. He also informed me that no one is ultimately responsible for their actions, yet we can proximally respond to our circumstances with intelligence. Given the chance can learn from our mistakes, right our wrongs, send healthy vibrations in all directions and merge into Infinity with a wave that perpetually benefits future generations. Everyone can potentially be a genius!”
Tell me why this works?
To solve this problem using a system of equations, we need to define variables for the number of chickens and cows that the first farmer purchased.
Let's say:
- x: Number of chickens purchased
- y: Number of cows purchased
Now, let's translate the information given into equations.
Equation 1: The total number of heads (chickens + cows) is 30.
x + y = 30
Equation 2: The total number of feet (2 legs for chickens and 4 legs for cows) is 100.
2x + 4y = 100
Now, we have a system of two equations:
Equation 1: x + y = 30
Equation 2: 2x + 4y = 100
We can solve this system of equations using different methods, such as substitution, elimination, or graphing. Let's solve it using the substitution method:
Step 1: Solve Equation 1 for x:
x = 30 - y
Step 2: Substitute the value of x in Equation 2:
2(30 - y) + 4y = 100
Step 3: Simplify and solve for y:
60 - 2y + 4y = 100
2y = 100 - 60
2y = 40
y = 40/2
y = 20
Step 4: Substitute the value of y back into Equation 1 to find x:
x + 20 = 30
x = 30 - 20
x = 10
Therefore, the first farmer purchased 10 chickens and 20 cows at the market.