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Systems of Equations: Solving using Elimination and Substitution.

Coming across multiple equations, both containing multiple unknown values, a common way to solve these problems are using methods called Elimination and Substitution. Depending on the type of problem, one method may be easier than the other, but both methods will ultimately find the correct answer.

The method of substitution does exactly as described; it finds some value, and then you substitute that value in for all cases of the variable that you solved.

Let's try one:

Ex. 1: Solve:
-2X + 3Y = 11
 2X + Y = 9

Step 1: Solve for one of the unknown variables, X or Y. It's best to choose whichever variable is not being multiplied by other numbers, so in this problem, we still solve for Y in the second equation:

Y = 9 - 2X

Now that we have our value for Y, we can substitute into the first equation:

-2X + 3(9 - 2X) = 11
-2X + 27 - 6X = 11
27 - 8X = 11
-8X = -16
X = 2

Now that we have X, we can substitute that value back in to either equation to solve for Y.

2(2) + Y = 9
4 + Y = 9
Y = 5

So when using substitution, all you need to do is use one of the equations to get a value - then plug it into either of the original equations to get the other unknown value!


Solving Using Elimination
There are some problems that using elimination is a little quicker to solve the problems. If the two equations contain a same value for X or Y, elimination may be the way to go:

X - 2Y = 14
X + 3Y = 9

Since both X values are the same (they are both "1X"), we can use them. Here's how:

Step 1: Subtract the two equations, straight down. This effectively eliminates the X variable:

  X - 2Y = 14
  X + 3Y = 9
     -5Y = 5

Since we're subtracting the entire line, the X's are eliminated. We end up with -5Y since -2Y - (3Y) is -5Y. And 14-9, is 5.

So Y = -1. Using this, we can put the -1 into either equation to solve for X:

X - 2(-1) = 14
X + 2 = 14

X = 12


The most common mistake happens when, during the first step, the values are subtracted. Remember to use your rules for adding and subtracting numbers (and negative numbers) when eliminating values!

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