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Algebra for Adults:
Part 7, The Order of Operations:
January 13, 2003:
First the challenges from last time.
Express each one of these algebraically.
75 subtracted from the quotient of b divided by c.
b/c - 75.
The sum of a and b divided by the product of c and d.
a + b/cd
A length (x) added to three times a length (y) divided by 2.
x + 3y/2
A speed of twice x diminished by a speed of y divided by c.
2x - y/c
The product of x and y and z added to the quotient of c divided
by 2.
xyz + c/2
Half of z increased by a times 35.
Z/2 + 35a
If you had a problem with any of these you may want to review Parts 5 and 6.
Now we can talk about the order of evaluation of expressions. Evaluation is a formal math word. It just means finding the answer or solution.
Normally, when the order of evaluation of an expression may
not be clear, we use parentheses to make it clear.
For example:
Evaluate the expression 3 + 4 x 6.
Notice that the answer will be different, depending on the order
of evaluation we choose.
3 + 4 x 6 = 7 x 6 = 42
or
3 + 4 x 6 = 3 + 24 = 27
For sure, this will not do.
To make it clear we will use parentheses to show how to proceed.
For example:
Evaluate the expression (3 + 4) x 6.
Or
Evaluate the expression 3 + (4 x 6).
The rule is, we always evaluate the part of the expression enclosed in parentheses first.
So, (3 + 4) x 6 = 42
and
3 + (4 x 6) = 27
Most of the time, we can avoid ambiguity through the use of
parentheses.
However, when we encounter an expression without parentheses we
have a procedural rule.
To evaluate and expression without parentheses, do the multiplication and division in order from left to right. Then do the addition and subtraction in order from left to right.
For example:
Evaluate the expression 6 + 5 x 4/2 - 4 + 2 x 7.
First do the multiplication and division in order from left
to right.
6 + 5 x 4/2 - 4 + 2 x 7 = 6 + 20/2 - 4 + 14 = 6 + 10 - 4 + 14
Then do the addition and subtraction in order from left to
right.
6 + 10 - 4 + 14 = 26
We did the above examples using just numbers. In algebraic
expressions where we use letters to represent numbers the rules
still apply.
For example:
Evaluate the expression 7(z + y) + 2z - (y +3)/2 where z = 3 and y = 5.
First we substitute our values for the letters.
7(3 + 5) + 2 x 3 - (5 + 3)/2
Then we evaluate the parts within the parentheses.
7 x 8 + 2 x 3 - 8/2
Then we do the multiplication and division from left to right.
56 + 6 - 4
Finally we do the addition and subtraction from left to right.
56 + 6 - 4 = 58
Okay, that's enough for now.
Here are a few practice expressions.
Evaluate the following:
3(4 - 2 + 8) + 2(a - c)/c where a = 5 and c = 2
6(a + b/2) - 2(b - c)/c where a = 6, b = 8, and c = 3
a(2b + 4c)/(a - 2) where a = 4, b = 2, and c = 3
x(37y - 22z)/x = 4 where x = 4, y = 5, and z = 257
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