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Solving Systems of Linear Equations in Three Variables, a la Shmoop.

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James, Clara, and Riley want to start a petting zoo together...with pandas, chameleons, and

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macaws. They each come up with a separate plan for

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the number of animals they want to buy... ...but they aren't exactly sure how much each

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individual animal costs. James realizes that if he buys 4 pandas, 7

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chameleons, and 10 macaws, it'll cost him $104.

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Clara finds that if she buys 2 pandas, one chameleon, and sells 3 macaws, she'll have

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paid $13.

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Riley figures out that if she buys 5 chameleons and one macaw, she'll have paid $33.

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What is the price of each animal? First, let's define our variables. We can

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label "p" as the price of one panda, "c" as the price of a chameleon, and "m" as the price

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of a macaw. Next, let's write equations for what we know.

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Starting with James, we know that if he buys 4 pandas, 7 chameleons, and 10 macaws, it'll

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cost him $104.

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Because p represents the price of one panda and 4 represents the number of pandas, we

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can write the cost of 4 pandas as 4p.

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We can write the cost of 7 chameleons as 7c, and 10 macaws as 10m.

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We know total cost is 104 dollars, so 4p plus 7c plus 10 m equals 104.

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We can write similar equations for Clara and Riley.

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For Clara, we have 2p plus c minus 3m equals 13, and 5c plus m equals 33.

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To solve this system of linear equations of three variables, we will first use elimination

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to reduce the system into two variables.

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Let's look at the top two equations first. Since we don't have equations where we can

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eliminate a variable by addition or subtraction directly...

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...we can multiply one or both of the equations by a constant so that we CAN eliminate one

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of the variables by adding or subtracting. We want to eliminate p from the top two equations

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first because there is no "p" in the bottom equation.

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To do this, we can multiply the second equation by 2 to get 2p times 2 is 4p, plus c times

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2 is 2c, minus 3m times 2 is 6m, which equals 13 times 2..which is 26.

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Subtracting the first two equations, we get...5c plus 16m equals 78.

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Looking at the second and third equations together...

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...we can see that we now have two equations with two variables, so we can use elimination

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again.

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5c and 5c eliminate directly, so we can just subtract the two equations to get 15m equals

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45.

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Dividing both sides by 15, we see that m equals 3. So the price of one macaw is 3 dollars.

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Plugging "m" as 3 back into the last equation, we have that 5c plus 3 equals 33.

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Subtract 3 from both sides, to get 5c equals 30.

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Divide 5 from both sides, and we get c equals 6.

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So... the price of one chameleon is 6 dollars. Finally, to solve for the price of one panda,

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we can substitute "m" as 3 and "c" as 6 back into either the first or second equation.

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Using the first equation, 4p plus 7 times 6 plus 10 times 3 equals 104.

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7 times 6 equals 42 and 10 times 3 equals 30. So 4p plus 42 plus 30 equals 104.

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Subtracting 42 and 30 from both sides, 4p equals 32.

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Divide both sides by 4... and p equals 8 dollars. Whew! We've found that one panda is 8 dollars,

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one chameleon is 6 dollars, and one macaw is 3 dollars.

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Pretty reasonable prices for such exotic animals...

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...but it probably still won't save James, Clara and Riley enough money to make up for

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all the lawsuits.

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