Solving Linear Equations

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So, you want to solve a linear equation?

Well, first there are some terms you'll have to know (trust me on this, my teacher gives us loads of vocabulary, and I am thankful!)

Linear Equation: equation with no exponents on the variables greater than 1, only one solution, and when graphed forms a straight line

Examples:
x + 3 = 10, 2[x - 9(x + 3)] + x = 182, x + y = 3, -x - 7|y| = 15

**Note: At this time, stay away from using "i" as a variable (also "o" because it looks like a 0)

Solution: the value of the variable that makes the equation true

OK....

Now I must introduce a VERY IMPORTANT PRINCIPLE that you will use in ALL equations!!!!!!

****Balance****

Repeat after me, make it your mantra: "What you do to one side, you must do to the other."

Well now let's try one together, a simple one.

x + 3 = 10

Now, we know what x + 3 equals, but what we want to know is what x equals.  To do that, we want to get rid of the 3 from the left side.  To do that, we add the opposite (change the sign...+ becomes -, - becomes +) of it. This is -3.  REMEMBER: We add -3 to the left side, but also the right side!!!!!!!!

x + 3 = 10

+    -3   -3  
x = 7

Say we try another one...
x + 5 + x = 15
Now we must combine like terms (as you already learned from another tutorial page, hint hint).

So we have 2x + 5 = 15

Now we add -5 to both sides:

2x + 5 = 15
+    -5    -5
2x = 10

Now we cannot simply add or subtract to find the solution.  We must "multiply by the reciprocal of the coefficient of the variable," or basically divide by the coefficient.  For example, the coefficient of x is 2.  The reciprocal of 2 is 1/2.  So, we multiply both sides by 1/2.  Make sure that if for some reason you have more than one term on the other side, you do it to each one.

(1/2) * 2x = 10 * (1/2)
x = 5

Summary of Steps to Solving a Linear Equation
1. (Clear all parentheses, fractions and decimals.)
2. Combine like terms.
3. Add the opposite to get the variable (and its coefficient) alone on one side.
4. Multiply both sides by the reciprocal of the coefficient of the variable to get the solution.

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