College Mathematics NotesSection 2.1Page 1 of 5
Chapter 2: Algebra Basics
Section 2.1: Solving Linear Equations in One Variable
Big Idea for this section:An equation is a statement that two expressions are equal to each other. The goal of this section is to learn how to solve linear equations. This involves performing the same arithmetic operation to both sides of the equation.
Big Skills for this section: You should be able to solve linear equations.
Section 2.1.1 Introduction to the Concept of an Equation
In the last section, we learned how to write a single expression to represent a multi-step calculation.
For example, suppose you are planning a vacation, and you know it will cost $100 in gas to get there, the hotel room is $150 per night, and you plan on spending $50 per day on food. If it takes about a day to drive to your destination, then you will have one less night in the hotel than the number of days of your vacation because you’ll spend the last day driving. If you use the variable N to represent the number of days you have for your vacation, then an expression for the cost is:
$100 + ($150)(N – 1) + ($50)N + $100
If you knew you had N = 5 days for your vacation, then you could evaluate the expression to find out the total cost of your vacation:
$100 + ($150)(N – 1) + ($50)N+ $100
= $100 + ($150)(5 – 1) + ($50)5 + $100
= $100 + ($150)(4) + ($50)5 + $100
= $100 + $600 + ($50)5 + $100
= $100 + $600 + $250 + $100
= $700 + $350
= $1,050
Now let’s say that you have $1,850 to spend on your vacation (that’d be nice, right?). How can you figure out how many days you can afford? From the calculation above, you know that it is more than five days. You could just keep trying bigger and bigger numbers for N until you hit $1,850. Or you could write down and then solve the following equation.
$100 + ($150)(N – 1) + ($50)N + $100 = $1,850
An equation is a mathematical statement where two algebraic expressions are set equal to each other.
Equations are used when we know what we want the answer to a calculation to turn out to be, but we don’t know what number to put in the calculation to get that answer. In the example above, we know we want the answer to be $1,850, but we don’t know how many days to put into our expression to get that answer.
In this section we will learn how to solve basic equations. Solving an equation simply means that you figure out what number the variable has to be to make both sides of the equation evaluate to the same number.
The value of the variable that makes both sides of the equation evaluate to the same number is called a solution to the equation.
Some equations have more than one solution. We will not learn about those equations in this class. We will focus on “linear equations”, which will always have just one solution. Linear equations have a variable but no exponents.
Section 2.1.2 Solving Equations Using the Addition Property of Equality
To solve an equation, you add, subtract, multiply, or divide both sides of the equation by the same number until you end up with an equation that says:
variable = a number.
This process is called isolating the variable, and notice that it is a statement of the solution to the equation.
In the vacation example above, if you were to correctly solve the equation, you would come up with the following statement. We will build up our knowledge of how to get that solution in pieces.
N = 9
The first piece of knowledge you need to solve an equation is the addition property of equality.
The addition property of equalitystates that you can add or subtract the same number from both sides of an equation without changing the fact that both sides remain equal.
Examples:
3 = 32(9-6) = 6
3 + 5 = 3 + 52(9-6) + 10 = 6 + 10
To use the addition property of equality to solve an equation, either add or subtract a number from both sides of the equation to make a zero on the side of the equation with the variable. Usually, the equation gets simpler-looking if you apply the property correctly. The simpler-looking equation is called an equivalent equation, because the value of the variable that makes it true is the same as the value that makes the harder equation be true.
Practice:Solve x – 5 = 4
Even though you likely got the solution of x = 9 without thinking too hard, you should practice showing how to use the addition property of equality to get that solution:
You should always check your answers after solving an equation by putting your solution back into the original equation, and then verifying that both sides evaluate to the same number.
Practice:Solve u + 19 = 7
The second piece of knowledge you need to solve an equation is that sometimes you’ll need to simplify expressions in the equation before you use the addition property of equality. There are two simple ways to simplify:
(1) Use the distributive law (i.e., multiply through any parentheses).
(2) Combine like terms on each side of the equation.
Practice:
Solve 7y – 2(3y + 4) = 9
Solve 13y – 6(2y– 5) = 22
Section 2.1.3 Solving Equations Using the Multiplication Property of Equality
The third piece of knowledge you need to solve an equation is the multiplication property of equality.
The multiplication property of equalitystates that you can multiply or divide both sides of an equation by the same number without changing the fact that both sides remain equal.
Examples:
3 = 32(9-6) = 6
3 5 = 3 52(9-6) 2 = 6 2
To use the multiplicationproperty of equality to solve an equation, either multiplyor divide both sides of the equation by the same number to make a one as a factor in front of the variable. Note: you usually only use this property when you have a number times the variable alone on one side of the equation.
Practice:Solve the equation 5c = 40.
Practice:Solve 3x = –30.
Practice:Solve 1.4k = 4.06
Practice: Solve
Section 2.1.4 Solving Equations That Require Simplification
Procedure for solving any linear equation
1) First simplify each side of the equation as much as possible
a) Use the distributive property to remove parentheses groups
b) Combine like terms on each side.
2) Use the addition/subtraction properties of equality to move like terms to the same side of the equation. Your equation will have all variables on one side and all numbers on the other side at this point. Combine those like terms.
3)Use the multiplication property of equality to remove any remaining coefficient on the variable term by dividing both sides by the coefficient.
4) Check all solutions in the original equation to verify solutions. Correct for possible mistakes if your answers do not satisfy the equation.
Practice:Solve
Practice:Solve
Practice:Solve$100 + ($150)(N – 1) + ($50)N + $100 = $1,850