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Systems of equations can be amazingly useful. How else would you figure out how many ounces of 70% dark chocolate and 20% milk chocolate you need to mix to get one pound (16 oz) of 40% chocolate? We forgot to mention: the meaning of life involves mixing chocolate. There. Now you know.
While problems with two unknowns may seem impossible to solve at first, we can use a system of equations to organize our information, then use one of our three methods to solve the system and find the answer. Oh, organization. You're a cure for all ills.
This idea becomes even more useful when we allow the systems of equations to have more unknowns and more equations. Then we can solve for 3 things at once. Or 4, or 5, or 6. Not 7, though. Certainly not 7. All right, let's throw 7 into the mix.
Systems of equations can also be represented using matrices. A matrix is a rectangular grid full of numbers, like this guy:
The great thing about a matrix is that it keeps all those floating numbers trapped inside a box, so they can't escape. It might need to plug up a couple of holes at the top and bottom, but it's better than nothing.
The word "matrix'' has many interesting meanings. There's a movie called The Matrix. (We're not counting the "sequels," and you can't make us.) There's also a matrix in the science fiction television series Dr. Who. There was a character named "Dot Matrix" in the movie Spaceballs. We could go on, but we're getting tired of all the linking.
Anyway, back to math. The system of equations
could also be written as a matrix equation:
The subject of linear algebra deals with ways to solve matrix equations, as well as various other things relating to matrices. For example, if you get married one day, you may want to have a Matrix of Honor by your side.
There are three steps to solving a math problem.
Let's see this in action, shall we?
Samantha wants to buy some bags of nuts and some bags of raisins to make trail mix. Nuts come in a 3 oz bag, and raisins come in a 4 oz bag. Nuts cost $0.50 per ounce, and raisins cost $0.75 per ounce. Samantha wants two pounds of trail mix that will cost $21 total. How many bags each of nuts and raisins should Samantha buy? Never mind the fact that she's totally forgotten to add the M&M's.
1. Figure out what the problem is asking.
This step, in which we translate from English to math, is the hardest part. We want to find how many bags of nuts and how many bags of raisins Samantha will need to buy. We'd like to figure this out before Trader Joe's closes, so chop chop.
The numbers of bags are two unknowns, so let's have:
We want to find x and y given a whole bunch of other information. We need to organize that other info into two pieces, also known as a system of two equations.
The two things mentioned in the problem are weight ("Samantha wants two pounds of trail mix'') and cost ("that will cost $21''). We should have one equation talking about weight and one equation talking about cost. If you want, we can have a third equation talking about celebrity fashion, if you think it'll make the rest of the medicine go down easier.
Our first piece of information deals with weight. If Samantha buys x bags of nuts and y bags of raisins, how much will everything weigh? A bag of nuts weighs 3 oz and a bag of raisins weighs 4 oz, so the total weight, in ounces, is
3x + 4y.
We want two pounds of trail mix, or 32 ounces, so we want to have
3x + 4y = 32.
Our second piece of information deals with cost. A bag of nuts is 3oz of nuts at $0.50 per oz, so a bag of nuts costs $1.50. A bag of raisins is 4 oz at $0.75 per oz, so a bag of raisins costs $3. The total cost when Samantha buys x bags of nuts and y bags of raisins is
1.5x + 3y.
In order for this to cost $21, we must have
1.5x + 3y = 21.
We have two unknowns and two equations now. If x is the number of bags of nuts and y is the number of bags of raisins Samantha buys, then we want to solve the system of equations
for x and y.
2. Solve the problem.
To solve the problem, we solve the system of equations
We can multiply the second equation by 10 to get rid of the decimal point. This gives us
15x + 30y = 210.
Now we can divide this equation by 5 to make the numbers smaller and easier, which gives us
3x + 6y = 42.
Hmm...let's solve these guys by addition/elimination. Any other method is fine, too, but this one looks like it'll be quickest. Multiply that first equation by -1 and add both equations together:
Now we've got:
6y – 4y = 42 – 32
Simplify a bit and solve for y:
2y = 10
y = 5
To find x, we can use either of the original equations. We'll use the first one since it doesn't have decimals, and decimals give us cold sweats. We know y = 5, so
Samantha needs 4 bags of nuts and 5 bags of raisins.
3. Check the answer.
Let's look up at the original problem and make sure the numbers we got make sense. With 4 bags of nuts and 5 bags of raisins, the weight will be
3(4) + 4(5) = 32,
which is indeed 2 pounds.
The cost of 4 bags of nuts and 5 bags of raisins will be
1.5(4) + 3(5) = 6 + 15
which is 21, exactly like it should be.
Update: We've received word that Samantha has now added both M&Ms and toffee pieces to her trail mix. She's really raisin the bar.