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Systems of Linear Equations
Systems of Linear Equations
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How to Solve a Math Problem

There are three steps to solving a math problem.

  1. Figure out what the problem is asking.
     
  2. Solve the problem.
     
  3. Check the answer.

Sample Problem

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 has 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 information 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 will 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.

From this, we subtract the first equation:

We find

2y = 10,

so

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

4(3) + 5(4) = 32,

which is indeed 2 pounds.

The cost of 4 bags of nuts and 5 bags of raisins will be

4(1.5) + 5(3) = 6 + 15

which is 21, exactly like it should be.

Update: We have received word that Samantha has now added both M&Ms and toffee pieces to her trail mix. She is really raisin the bar.

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