<u><em>Preliminaries</em></u>
There are some basic things you must know. These include ...
- how to add, subtract, multiply, and divide fractions
- how to add, subtract, multiply, and divide decimals, including whole numbers
- how to use the distributive property to eliminate parentheses
- how to use the distributive property to "collect terms"
<u><em>The One Rule you must never break</em></u>
- Whatever you do to one side of the equation, you must also do to the other side.
My teacher expressed this as "keep the equal sign sacred." This is what the properties of equality tell you: you can add, subtract, multiply, or divide by anything (except dividing by 0), as long as you do it to <em>both sides of the equation in their entirety</em>.
So, in the following, whenever we say "add <something>" or "divide by <something>" we mean that you do that to <em>both sides of the equation</em>.
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<em><u>Step-by-step</u></em>
After you have solved a few linear equations, you can find ways to simplify the steps. This set of steps <em>will always work</em>, but may end up being slightly more work than you absolutely must do.
1. <em>Subtract one side of the equation</em> (from both sides, according to the One Rule). This will give you an equation of the form: <expression> = 0. It can make life easier later if you can first identify the side of the equation with the smallest or most-negative coefficient of the variable. You would choose that side of the equation to subtract. The following steps will work either way.
2. <em>Simplify the expression</em> to the form:
<coefficient> × <variable> + <constant> = 0
If fractions or multiple layers of parentheses are involved, use what you know from the above preliminaries. Either or both of the constant or coefficient may be negative. They may have "literals" involved (letters that you treat as numbers), possibly including units, such as meters or dollars or degrees. When fractions are involved, multiplying the equation by the denominator can "clear the fractions."
3. <em>Divide by the coefficient</em>. This gets you the form:
<variable> + <constant> = 0
If the coefficient was not 1, the constant in this equation will be different from the constant in the one of Step 2.
4. <em>Subtract the constant</em>. Now, you will have your solution in the form:
<variable> = <constant>
Of course, this constant is the opposite of the one in Step 3. If there are literals or units involved, the "constant" here may contain literals or units. Don't be alarmed. All the rules of algebra apply to those, too.
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<em><u>Example</u></em>
Here's a somewhat complicated example problem we want to solve for x.
3(x +2) +4 = -(x +1)/6 -10 . . . . . . . given linear equation
We see the right side has a negative coefficient for x, while the coefficient on the left side is positive. We'll choose to subtract the right side.
3(x +2) +4 -(-(x +1)/6 -10) = 0 . . . . subtract right side (step 1)
3x +6 +4 +(x +1)/6 +10 = 0 . . . . . . eliminate outer parentheses (step 2)
3x +6 +4 + x/6 +1/6 +10 = 0 . . . . eliminate parentheses (step 2)
(3 1/6)x + (20 1/6) = 0 . . . . . . . . . collect terms (step 2)
x + (20 1/6)/(3 1/6) = 0 . . . . . . . . divide by the coefficient of x (step 3)
x + 121/19 = 0 . . . . . . . . . . . . . . . . simplify the fraction (step 3)
x = -121/19 . . . . . . . . . . . . . . . . . . add the opposite of the constant (step 4)
We can check this result (recommended) to make sure it is the solution. Substituting it into the original equation, we get ...
3(-121/19 +2) +4 = -(-121/19 +1)/6 -10
3(-83/19) +4 = (102/19)/6 -10
-249/19 +4 = 17/19 -10
-173/19 = -173/19 . . . . . . true