How to Solve Equations
Solving an equation means getting the variable (we'll use x) on one side
of the
equal sign all by itself. So if you have the equation
ax + b = c then solving it means getting the x on one
side of the equal sign by itself. ("a","b"', and "c" stand for real
numbers).
You can perform the same
operation (add, subtract, multiply, divide) on both sides of the
equation. So you could add 3 to both
sides like this:
ax + b = c
+ 3 + 3
-------------------
ax + b + 3 = c + 3
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This equation is true also, just like the first. But it didn't do you any good! The idea is
to get rid of terms on the
same side of the equation as x. Look carefully at the equation to see what
operation you can "undo". Then choose the opposite. So you would like to get rid of the "+b", by subtracting:
ax + b = c
- b = c - b
--------------------
ax = c - b
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Notice that the +b -b leaves nothing.
Now, since x is multiplied by a, you divide both sides of the
equation by a:
ax c - b
---- = -----
a a
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So the a's cancel and we have:
x = c - b   
        a |
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The general rules for doing this are:
- Always do an operation to BOTH sides of the equation
- Do adds and subtracts first, then multiplies and divides
- Always do the opposite operation to move the term to the
other side of the equation:
- If a term is added to x, then subtract it from both sides
Example: x + 5 = 15 ----> you subtract 5 from both sides
- If a term is multiplied by x, then divide both sides by it
Example: 3x = 15 ----> you divide both sides by 3
- If x is divided by a term, then multiply both sides by it
Example: x/2 = 20 ---> you multiply both sides by 2
Example 1:
3x + 5 = 17 <---- The original equation
3x + 5 = 17
- 5 -5 <---- Step 1: subtract 5 from both
------------ sides to remove it
3x = 12
3x = 12
--- --- <---- Step 2: divide both sides
3 3 by 3 to remove the 3
x = 4 <---- The answer!!! |
Example 2:
x/3 - 2 = 3 <---- The original equation
x/3 - 2 = 3
+ 2 +2 <---- Step 1: add 2 to both
---------------- sides to remove the -2
x/3 = 5
3x/3 = 3 x 5 <---- Step 2: multiply both sides
by 3 to remove the 3
x = 15 <---- The answer!!! (3's cancelled) |
Example 3:
35 - 3x = 14 <---- The original equation
35 - 3x = 14
-35 -35 <---- Step 1: subtract 35 from
------------------ both sides to remove it
-3x = -21
-3x = -21
---- ----
-3 -3 <---- Step 2: divide both sides by -3
to remove the -3
x = 7 <---- The answer!!!
(remember, a negative divided
by a negative is a positive) |
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