When I initially studied algebra in
high school, linear equations were one of the few algebraic expressions that
made sense to me. These equations create a straight line that can be graphed. To
understand linear equations, four very important properties of equality needed
to be understood first:
Addition Property:
|
For all real numbers x, y, and z,
if x = y, then x + z = y + z.
|
Subtraction Property:
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For all real numbers x, y, and z,
if x = y, then x – z = y – z.
|
Multiplication Property:
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For all real numbers x, y, and z,
if x = y, then xz = yz.
|
Division Property:
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For all real numbers x, y, and z,
if x = y, and z ≠ 0, then x/z = y/z.
|
In essence, whatever has to be done
to one side of the equation in order to isolate the unknown quality has to be
done to the other side of the equation to maintain equality. Here is an
example:
Solve the equation: 3x
+ 5 = 11
1. Isolate 3x on one side by
subtracting 5 from both sides using the subtraction property of equality.
3x + 5 – 5 = 11 - 5
3x
= 6
2. Next, divide both sides of the
equation by 3 to isolate the value for x. This step uses the division property
of equality.
3x/3
= 6/3
X = 2
3. The last step involves checking
you work by placing the value of x into the original equation.
3(2)
+ 5 = 11
6 + 5 = 11
The value of x (2) makes the statement
true. We have found our answer.
Knowing the
properties of equality enables us to solve more complex equations as well. When
thinking about these properties and their relation to linear equations, I
picture two platters that need to remain equal. In looking at the equation
above, if I take 5 cookies off of the first platter, I need to take 5 cookies
off of the second platter to keep them equal. If I divide one platter by 3 to
reduce the amount of cookies further, I need to do the same to the second
platter to keep them identical.
Here is another look at solving
linear equations:
When solving
linear equations, remember that the properties of equality are essential in
finding the unknown value. Equality matters! Here is a great game that can be
used to practice solving linear equations:
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