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Explanation:
The rules of equality tell you that you can do anything you like to an equation, as long as you do the same thing to both sides of the equal sign. This is the essence of Algebra, so is worth remembering. My teacher summarized it as "keep the equal sign sacred."
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"Simple linear equations" come in several varieties. They are sometimes called "one-step", or "two-step", or "three step" linear equations, depending on how many steps are required to solve them. In what follows, we're using "x" as the variable, and the letters a, b, c, d as constants.
<u>One-step</u>
In general, the "one step" will be either an addition or a multiplication.
x + a = b . . . . an addition equation
ax = b . . . . . . a multiplication equation
The key in any case is to look at what is done to the variable and use an inverse operation to "undo" it. The opposite of addition is addition of the additive inverse (add the opposite). This is also called subtraction. For the addition equation, this looks like ...
x + a - a = b - a . . . . . . add (-a) <em>to both sides </em>(or subtract 'a')
x = b - a . . . . . . . . . . . the solution
The opposite of multiplication is multiplication by the multiplicative inverse (the reciprocal) This is also called division. For the multiplication equation, this looks like ...
ax(1/a) = b(1/a) . . . . . multiply by (1/a) <em>on both sides</em> (or divide by 'a')
x = b/a . . . . . . . . . . . the solution
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<u>Two-step</u>
As with one-step linear equations, two-step equations come in some different forms.
ax +b = c
ax +b = cx
a(x +b) = c
In general, the two steps are an addition step and a multiplication step, not necessarily in that order. You will notice that the first step can transform the equation to a "one-step" equation. As before, inverse operations are involved, and whatever is done is done to both sides of the equal sign.
ax +b = c ⇒ ax = (c -b) . . . . subtract b to get a multiplication equation
In this next, we have the variable on both sides of the equal sign. we can put the variable on one side of the equal sign by adding the opposite of the term we don't want where it is. Leaving the constant alone, we can add (-ax) to both sides to get ...
ax +b = cx ⇒ b = cx -ax ⇒ b = (c -a)x
When you have numbers, you can go directly to (c-a)x by "combining terms". This is now the one-step multiplication equation.
Finally, another 2-step form is ...
a(x +b) = c ⇒ x +b = c/a . . . . . divide by 'a' to get an addition equation
This last equation can also be solved by using the distributive property first.
ax +ab = c . . . . . still a 2-step equation, now of the first type
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<u>Three-step</u>
The most typical "3-step" linear equation is of the form ...
ax +b = cx +d
where both variables and constants are on both sides of the equal sign. As you may have noticed above, we prefer to have variables on one side and constants on the other side. So, add the opposite of the term you don't want where it is (to both sides, of course).
If we want variables on the left and constants on the right, we can add -cx and -b. The "3 steps" assume you do these as separate operations.
ax +b -cx = cx +d -cx ⇒ (a-c)x +b = d . . . a two-step equation
(a-c)x +b -b = d -b ⇒ (a-c)x = d-b . . . . a one-step multiplication equation
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<em>Additional comments</em>
There are some choices you can make that will tend to reduce errors. One of them involves selection of the variable term you want to eliminate (add the opposite of). Generally, if you choose the term with the lowest (most negative, left-most on the number line) coefficient, adding its opposite will result in a positive coefficient for the remaining variable term. Then when you divide by that coefficient, you're dividing by a positive number. Most folks perform arithmetic with fewer errors when they're working with positive numbers.
The general approach to any "solve for" situation is to look at what is done to the variable you're solving for. The Order of Operations is a useful lens for looking at this. When you list the operations, your "undo" sequence will start with the last operation on the list, and work backward.
<em>Further note</em>: your total comfort with doing arithmetic with any kind of numbers (integers, fractions, mixed-numbers, decimals, scientific notation, positive or negative) will aid you immensely. Always be careful with minus signs. They tend to get lost, forgotten, ignored, Be sure they don't.
