Algebra is a system of symbols and rules for manipulating those symbols to show mathematical processes in a shorthand notation. Functions use algebraic notation to show how an input number is changed to an output number by a series of arithmetic operations.
Expressions use algebraic notation to show a series of operations on a number. The instructions “take a number, multiply it by 7, and subtract 3” can be shown by the expression .
Using Arithmetic with Expressions
It is possible to do arithmetic with variable expressions by following certain rules. Expressions can be added or subtracted as long as the same variables are in each expression and raised to the same power.
works, but will not work because c has a different exponent in the 2 expressions.
Multiplication and division can be done, regardless of the exponents, by following rules for exponents.
Solving Problems with Expressions
Word problems can be turned into expressions that can be used to find a solution. For example, a clothing store sells hats for $4 and scarves for $3. What is their total revenue from these items if they sell 16 hats and 20 scarves? Use r to represent the total revenue. The revenue from each item is the number of items sold multiplied by the price per item.
Properties of Operations
There are certain properties, or “rules,” that govern how numbers work. Some of the most commonly used ones, and what they mean, are these:
Commutative Property: This works for addition and multiplication and simply says that it doesn’t matter what order two numbers are in when performing these operations—the answer will be the same. Examples: 2 + 3 = 3 + 2 and 4 x 6 = 6 x 4
Associative Property: This is the same as the commutative property, but refers to three or more numbers being multiplied or added. So (3 + 1) + 4 = 3 + (1 + 4) and (2 x 6) x 3 = 2 x (6 x 3). (Remember that operations in parentheses are performed before others. See “PEMDAS”, or “Order of Operations.”)
Neither of these properties apply to subtraction and division—number order does matter in those operations.
Distributive Property of Multiplication: This property states that you will get an equal result by multiplying a number by a whole or the parts of another number, such as: 3(2 + 4) = 3 x 2 + 3 x 4 = 18 and 3(2 + 4) = 3 x 6 = 18. In other words, you can “distribute” the multiplier over addition.
Some other properties with which you should be familiar are:
Order of Operations
When there is more than one type of operation to be done, the order of operations specifies which operations are done first, in order to get a correct answer. The order is abbreviated as PEMDAS, which can be remembered using “Please Excuse My Dear Aunt Sally.” The letters stand for:
Evaluate any expressions within Parentheses.
Evaluate any Exponents.
Evaluate all Multiplication and Division, from left to right.
Evaluate all Addition and Subtraction from left to right.
Parentheses are important because they can be used to force the operations to be done earlier in the sequence. For example, evaluate the expression:
The expression in parenthesis is evaluated first:
Exponents are evaluated next:
Multiplication and division is done from left to right:
Addition and subtraction is done from left to right:
The value of the expression is .
An equation is a statement that two expressions are equal. An inequality compares the expressions and states that one is greater than (), greater than or equal to(), less than(), or less than or equal to() the other expression. They are solved by performing the same operation on both sides of the sign. Multiplying or dividing an inequality by a negative number will cause the inequality sign to be reversed.
Equations and Inequalities in One Variable
Equations and inequalities in one variable are solved by performing the same operation on both sides until the variable is isolated on one side. The other side will be the solution. The operations on the equations are best done in the reverse of the order of operations.
To solve the equation :
Remove the “-7” by adding 7 to both sides:
(The equation remains valid but the variable is closer to being isolated.)
To isolate x, divide both sides by 4: .
The equation is now solved for the value of x.
To solve the inequality :
Subtract 12 from both sides:
Divide both sides by -3.
(Dividing by a negative number requires that the sign be reversed.)
The inequality is solved.
Notice that, in both cases, the best operation to do first was addition or subtraction of the constant term (term without the variable in it).
Linear equations can be reduced to the form or , where x and y are variables and m, a, b, and c are constants. When a linear equation is graphed on the coordinate plane, the result is a straight line. This represents a unique value of y corresponding to every value of x.
Pairs of Simultaneous Linear Equations
When 2 linear equations are given, there will be a single value of x and y that will be a solution for both equations, as long as the corresponding lines are not parallel. For example, find the solution for the system of equations and .
First solve one equation to get an expression for one of the variables in terms of the other:
Next, substitute this expression for x in the other equation:
Now, substitute the value of y into the expression derived from the first equation to solve for x. (It is best to use the improper fraction form of y to do the calculations.)
The solution for the system of equations is .
Graphic Representation of equations and Inequalities
Equations and inequalities can be plotted on a graph. Linear equations are straight lines, and inequalities are shaded regions on one side of a line.
If the inequality is or , the line is not part of the solution.
For the inequalities and , the line is part of the solution.
When a pair of equations or inequalities is graphed, the point of intersection or the region of overlap of the inequalities shows the solution of the system. This example shows the solution for the inequalities and .
A function is an equation that specifies a value of an output variable (usually y) for each value of an input variable (usually x). Some examples are , , and .
Interpreting a Function
A function gives us a relationship between the variables given in the function. The relationship can be read directly from the function or from the graph of the function. For example, in the function the -3 coefficent of x tells us that the value of y decreases as x increases. This can be seen on the graph as a line sloping down to the right.
Building a Function
A function can be constructed from information given in a problem. For example, suppose a museum charges a base fee of $50 plus $3 per student for classes to visit the museum. Let c be the total cost of the trip and s be the number of students. A function describing the cost of the trip for a given number of students is .