Section 2.2 Solving Linear Equations 117 3. Conditional Equations, Identities, and Contradictions The solutions to a linear equation are the values of x that make the equation a true statement.A linear equation in one variable has one unique solution. Some types of equations, however, have no solution while others have infinitely many solutions. I. Conditional Equations An equation that is true for some values of the variable but false for other values is called a conditional equation. The equation for example, is true on the condition that x 2. For other values of x, the statement x 4 6 is false. x 4 6 x 4 4 6 4 (Conditional equation) Solution set: {2} II. Contradictions Some equations have no solution, such as There is no value of x, that when increased by 1 will equal the same value increased by 2. If we tried to solve the equation by subtracting x from both sides, we get the contradiction This indicates that the equation has no solution. An equation that has no solution is called a contradiction.The solution set is the empty set.The empty set is the set with no elements and is denoted by { }. x 1 x 2 x x 1 x x 2 (Contradiction) Solution set: { } 1 2 III. Identities An equation that has all real numbers as its solution set is called an identity. For example, consider the equation, x 4 x 4. Because the left- and right-hand sides are identical, any real number substituted for x will result in equal quantities on both sides. If we subtract x from both sides of the equation, we get the 4 4. identity In such a case, the solution is the set of all real numbers. x 4 x 4 x x 4 x x 4 (Identity) Solution set: The set of real numbers. 4 4 Identifying Conditional Equations, Contradictions, and Identities Example 8 Solve the equation. Identify each equation as a conditional equation, a contradiction, or an identity. 4k 5 212k 32 1 21b 42 2b 7 3x 7 2x 5 a. b. c. Solution: a. Clear parentheses. Combine like terms. Subtract 4k from both sides. 4k 5 212k 32 1 4k 5 4k 6 1 4k 5 4k 5 4k 4k 5 4k 4k 5 5 5 1Identity2 This is an identity. Solution set:The set of real numbers. 1 2. x 1 x 2. x 2 x 4 6, TIP: The empty set is also called the null set and can be expressed by the symbol . Avoiding Mistakes There are two ways to express the empty set: { } or . Be sure that you do not use them together. TIP: In Example 8(a), we could have stopped at the step 4k 5 4k 5 because the expressions on the left and right are identical.
miller_beginning_intermediate_algebra_4e_ch1_3
To see the actual publication please follow the link above