Download Elementary Algebra Lecture Notes, Section 1.3

Elementary Algebra Notes
Section 1.3
Page 1 of 3
Section 1.3: Variables, Expressions, and Equations
Big Idea: An algebraic expression is a way of stating a multi-step arithmetic calculation where you want to
capture the steps of the calculation without necessarily using specific numbers, so you use letters (or variables)
to stand in for the specific numbers that may change depending on the situation. When we know what we want
the answer from a calculation to be, then we write the algebraic expression as being equal to the desired answer.
This is called an equation.
Big Skill: You should be able to translate a multi-step calculation into an algebraic expression, evaluate
algebraic expressions given a specific number for the variable, and verify that a number is a solution of an
equation.
A variable is a symbol, usually a letter like x, y, or z, used to represent any unknown number.
An algebraic expression is a sequence of numbers, variables, grouping symbols, and symbols of operations
(addition, subtraction, multiplication, division, or exponents) arranged according to the order of operations.
Algebraic expressions are meant to capture the steps of a multi-step calculation without being restricted to fixed
numbers all the time.
Examples:
x+5
2m – 9
8p2 + 6(p – 2)
Evaluating an algebraic expression means to plug a given number into the expression and calculate a
numerical answer.
Examples:
2m – 9 evaluated for m = 10 is: 2(10) – 9 = 20 – 9 = 11
Practice:
1. Write out the steps to calculate the total purchase price of a $37.99 shirt including 5.5% sales tax. Then
create an algebraic expression that describes how to calculate the total purchase price for any item
including the sales tax. Evaluate the expression for a $125.00 purchase.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 1.3
Page 2 of 3
2. Write out the steps to calculate the total purchase price of three of shirts purchased at a cost of $29.99
each and two pairs of pants purchased at a cost of $34.99 each. Then create an algebraic expression that
describes how to calculate the total purchase price for any number of shirts and pants at these prices.
Evaluate the expression for a purchase of four shirts and four pair of pants.
3. Suppose that one golf club has a yearly fee $300 and each round of golf you play costs $36, while
another golf club has no yearly fee, but each round of golf costs $40. Write an algebraic expression for
the cost of each club in terms of the number of games of golf played. Evaluate each expression for 60
golf games played and 80 golf games played.
4n  n 2
4. Evaluate the algebraic expression
for n = -5.
n2
5. Translate the English phrase a number x subtracted from 48 into an algebraic expression.
6. Translate the English phrase the product of 6 and a number y into an algebraic expression.
7. Translate the English phrase twice the sum of a number x and 5 into an algebraic expression.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 1.3
Page 3 of 3
Equations:
We write an equation when we know what we want the answer to a multi-step to calculation to
turn out to be.
A solution to an equation is a number that we can “plug into” each side of the equation and have
both sides be equal. If we are given a possible solution to an equation, all we have to do is plug it into the
equation and see if both sides are equal to determine if it truly a solution or not.
Practice:
1. The cover charge at a comedy club is $10 per person, and sodas cost $1.50 each once you’re in the club.
Imagine that you are going to pay for a friend and yourself to go to the club. Write an algebraic
expression that describes the cost in terms of how many sodas are purchased. Then write an equation
that describes how many sodas you can buy if you only have $35 to spend. Check to see whether 8, 9,
or 10 sodas is a solution to the equation.
2. For the golf club example, write an equation that describes the situation where the two clubs cost the
same amount of money. Then determine if 70 or 75 games played is a solution.
3. Determine if x = 1 is a solution of 5x + 12 = 7 – 3x.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.