Back to Algebra for
Adults.
Algebra for Adults:
Part 9, First Review solutions:
February 3, 2003:
In arithmetic, we found there were just four basic operations to consider. These being, Addition, Subtraction, Multiplication, and Division.
The other things we studied, Fractions, Decimals, and Percents, turned out to be different ways of looking at mathematical situations but the four basic operations applied to all of them.
The big difference in algebra is that we will represent numbers
with symbols, namely characters of the alphabet.
This will make any mathematical statement much shorter that its
verbal equivalent. In turn, that will make the statement easier
to understand and manipulate.
A verbal statement, might be: five times any number plus two
times the same number is equal to seven times the number.
Let the letter n stand for the number and write the equivalent
algebraic statement.
5n + 2n = 7n.
In algebra:
When we multiply a number by a letter the multiplication sign
may be omitted.
The commutative law of addition is:
Interchanging addends does not change their sum.
The Commutative law of multiplication is:
Interchanging factors does not change their product.
a + b = c and b + a = c are equivalent statements.
By the commutative law of multiplication, ab = c and ba = c are equivalent statements.
Microsoft Bookshelf gives this definition:
Equation
Mathematics: A statement asserting the equality of two expressions,
usually written as a linear array of symbols that are separated
into left and right sides and joined by an equal sign.
This shorthand (equation) way of representing mathematical relationships is the big difference between basic mathematics and algebra.
It is important to notice that algebraic expressions need not
be complete equations.
For example:
3x is an algebraic expression that means 3 times x.
12/y is an algebraic expression that means 12 divided by y.
Division by zero is undefined or impossible in math. This is still true in algebra. In the case of 12/y above where y = 0 the solution is undefined.
Express a weight that is eight times larger than another weight
(w).
The answer is 8w.
Express a height that is 3 feet shorter than a height z.
The answer is z - 3.
Six times the sum of z and y is6(z + y).
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.
The rule is, we always evaluate the part of the expression enclosed in parentheses first.
Evaluate:
(3 + 4) x 6 = 42
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.
Evaluate the expression 7(z + y) + 2z - (y +3)/2 where z = 3 and y = 5.
7(3 + 5) + 2 x 3 - (5 + 3)/2
7 x 8 + 2 x 3 - 8/2
56 + 6 - 4
56 + 6 - 4 = 58
Here are some definitions that are used in algebra.
A term is a number or the product of numbers.
A factor of a term is any of the numbers multiplied to form the term.
We can have complex factors.
For example:
In cd(a + b), (a + b) is a complex factor.
The factors of this expression are c, d, and (a + b).
We can also have more than one term in an expression.
For example:
In the expression b + bd - (b - d) the terms are b, bd, and (b
- d).
Any factor or group of factors of a term is a coefficient of the product of the remaining terms.
In the expression 6(x + y) 6 is a coefficient of (x + y) and (x + y) is a coefficient of 6.
We have literal coefficients and numerical coefficients.
In the expression 5xy
5 is the numeric coefficient of xy?
xy is the literal coefficient of 5.
In basic math, we learned to raise a number by a power or multiply
it by itself a number of times. We indicate that operation with
a superscript number.
For example:
Formally:
We call the superscript number the exponent.
We call the number being multiplied the base.
The result of the operation is called a power of the base.
For example:
125 is the third power of 5.
5 is the base.
3 is the exponent.
In certain case we can combine terms.
We can do this if the terms have identical literal factors and
exponents.
For example:
We can combine 8 + 7 + 2 into a single term 17.
It is not quite as obvious when we add literal factors to the
mix.
For example:
We can combine 12x + 8x into a single term 20x.
Back to Algebra for Adults.
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