## Properties of Equations
General Explanation
An EQUATION is a statement which shows that two quantities are equal.
Here are some examples of equations: 2 = 7 - 5 3x = 9xAlthough equations may contain only constants, we are mainly interested in those that contain variables, such as the second and third equations above. To SOLVE an equation is to find all the values of the variables for which the equation is TRUE. 2x = 8Substituting x = 4 makes the equation true, so 4 is a solution of the equation. All of the values for which an equation is true are known collectively as the equation's SOLUTION SET. A solution set is often enclosed in brackets. For example, the solution set of 2x = 8 would be written as {4}. Equations which have the same solution set are considered EQUIVALENT. It is not always easy to decide whether two equations are equivalent. However, by manipulating the equation we can always determine equivalence. The basic rules for manipulating equations correspond to the fundamental arithmetic operations: addition, subtraction, multiplication, and division. All of the rules stem from a single concept: Every time an arithmetic operation is performed on one side of the equation, the same operation should be performed on the other side. Here are the basic rules for solving equations: - The same value can be added to both sides of an equation without altering its solution set.
- The same value can be subtracted from both sides of an equation without altering its solution set.
- Both sides of the equation can be multiplied by the same value without altering its solution set.
- Both sides of the equation can be divided by the same value without altering its solution set.
Sample Problem 1
Determine if these two equations are equivalent:
#1: a + 9 = 11 #2: a + 5 = 7 Solution
Let's alter equation 2 to see if it is equivalent to equation 1. First we'll try
adding 4 to each side of the equation:
a + 5 + 4 = 7 + 4 a + 9 = 11Since we were able to alter equation 2 to arrive at a new equation identical to equation 1, the two equations are equivalent. Sample Problem 2
Determine if the following equations are equivalent:
#1: 35w = 70 #2: 7w = 14 Solution
Let's alter equation 2 to see if it is equivalent to equation 1.
We will multiply each side by 5:
7w • 5 = 14 • 5 35w = 70Since we were able to alter equation 2 to arrive at a new equation identical to equation 1, the two equations are equivalent. |