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Columbus State Community College
Chapter 2 Section 3 Solving Equations Using Addition

Solving Equations Using Addition
Determine whether a given number is a solution of an equation. Solve equations, using the addition property of equality. Simplify equations before using the addition property of equality.

Note on Identifying Equations
An equation has an equal sign. Notice the similarity in the words equation and equal. An expression does not have an equal sign.

Balanced and Unbalanced Equations
These equations balance, so we can use the = sign. These equations do not balance, so we write ≠ to mean “not equal to.” ≠ 14 = 12 = 3 • 5 24 – 8 ≠ 3 • 2

The Solution of an Equation
NOTE Most of the equations that you will solve in this book have only one solution, that is, one number that makes the equation balance. There are some equations that have two or more solutions. We will examine such equations later in this course.

Identifying the Solution of an Equation
EXAMPLE Identifying the Solution of an Equation Which of these numbers, 65, 85, or 75, is the solution of the equation m – 45 = 30? Replace m with each of the numbers. The one that makes the equation balance is the solution. Does not balance: 85 – 45 is 40 and 40 is more than 30. Does not balance: 65 – 45 is 20 and 20 is less than 30. Balances: 75 – 45 is 30. 65 – 45 ≠ 30 75 – 45 = 30 85 – 45 ≠ 30 The solution is 75 because, when m is 75, the equation balances.

Addition Property of Equality
If a = b, then a + c = b + c and a – c = b – c. In other words, you may add the same number to both sides of an equation and you may also subtract the same number from both sides of an equation and still keep it balanced.

A Real-World Example: Addition Property of Equality
If Ali and Ty each give away \$50, they will still have the same amount of money. If Ali and Ty are each given \$10, they will still have the same amount of money. If you add the same number to both sides or if you subtract the same number from both sides, the sides will remain equal. Ali’s Money Ty’s Money

Goal in Solving an Equation
The goal is to end up with the variable (letter) on one side of the equal sign balancing a number on the other side. We work on the original equation until we get: variable = number number = variable or Once we have arrived at that point, the number balancing the variable is the solution to the original equation.

Using the Addition Property of Equality
EXAMPLE Using the Addition Property of Equality Solve each equation and check the solution. (a) n = 32 n = 32 – – 7 n = 25 Solution Check the solution: n = 32 = 32 32 = 32 Balance statement

Using the Addition Property of Equality
EXAMPLE Using the Addition Property of Equality Solve each equation and check the solution. (b) x – 3 = –15 x – 3 = –15 x = –12 Solution Check the solution: x – 3 = –15 –12 – 3 = –15 –15 = –15 Balance statement

CAUTION CAUTION When checking the solution to Example 2(b), we end up with –15 = –15. Notice that –15 is not the solution. The solution is –12, the number used to replace x in the original equation. x – 3 = –15 x = –12 = –15 –15 –12 – 3 Solution Balance statement

Simplifying before Solving Equations
EXAMPLE Simplifying before Solving Equations Solve each equation and check the solution. (a) c – 4 = 9 – 5 c – 4 = 9 – 5 c – 4 = 4 c = 8 Solution Check: c – 4 = 9 – 5 8 – 4 = 4 4 = 4 Balance statement

Simplifying before Solving Equations
EXAMPLE Simplifying before Solving Equations Solve each equation and check the solution. (b) 2v – v = 3 – 2 2v – v = 3 – 2 v = 1 – 1 – 1 v = 0 Solution Check: 2v – v = 3 – 2 2 ( 0 ) – 0 = 1 1 = 1 Balance statement

Note on Eliminating Terms
To eliminate a term from one side of an equation, we perform the opposite operation that precedes the term we want to eliminate. Let’s take a another look at Example 2. n = 32 – – 7 n = 25 Example 2 (a) To get the variable, n, by itself we need to eliminate the constant term, 7, from the left side of the equation. The opposite of “+ 7” is “– 7”, so we subtract 7 from both sides of the equation.

Note on Eliminating Terms
To eliminate a term from one side of an equation, we perform the opposite operation that precedes the term we want to eliminate. Let’s take a another look at Example 2. x – 3 = –15 x = –12 Example 2 (b) To get the variable, x, by itself we need to eliminate the constant term, 3, from the left side of the equation. The opposite of “– 3” is “+ 3”, so we add 3 from both sides of the equation.

Note on Eliminating Terms
To eliminate a term from one side of an equation, we perform the opposite operation that precedes the term we want to eliminate. Let’s take a another look at Example 2. n = 32 – – 7 n = 25 Example 2 (a) x – 3 = –15 x = –12 Example 2 (b)

Solving Equations Using Addition
Chapter 2 Section 3 – Completed Written by John T. Wallace