6.1 Order of Operations and Solving Linear Equations 297 For example, suppose we want to evaluate the expression + x 2 3 when = x 4. Substituting 4 for x, we obtain + ⋅ 2 3 4. What is the value of + ⋅ 2 3 4? Does it equal 20, or does it equal 14? In mathematics, unless parentheses indicate otherwise, always perform multiplication before addition. Thus, the correct answer is 14. + ⋅ = + ⋅ = + = 234 2(34) 212 14 The order of operations for evaluating an expression is repeated here for your convenience. Timely Tip Some students use the phrase “ P lease E xcuse M y D ear A unt S ally” or the word “PEMDAS” ( P arentheses, E xponents, M ultiplication, D ivision, A ddition, S ubtraction) to remind them of the order of operations. Remember: Multiplication and division are performed left to right, and addition and subtraction are performed left to right. ORDER OF OPERATIONS 1. First, perform all operations within parentheses or other grouping symbols (according to the following order). 2. Next, perform all exponential operations (that is, raising to powers or determining roots). 3. Next, perform all multiplications and divisions from left to right. 4. Finally, perform all additions and subtractions from left to right. PROCEDURE Example 2 Evaluating an Expression Evaluate the expression − + + x x3 10 2 for = x 2. Solution Substitute 2 for each x and use the order of operations to evaluate the expression. − + + x x3 10 2 = − + + = − + + = + = (2) 3(2) 10 4 6 10 2 10 12 2 7 Now try Exercise 35 Solving Equations Two algebraic expressions joined by an equal sign form an equation . Some examples of equations are + = + = + = x x x x 24, 341, and 3 2 The solution to an equation is the number or numbers that replace the variable to make the equation a true statement. For example, the solution to the equation + = x 3 4 is 1. When we determine the solution to an equation, we solve the equation . We can determine if any number is a solution to an equation by checking the solution . To check the solution, we substitute the number for the variable in the equation. If the resulting statement is a true statement, that number is a solution to the equation. If the resulting statement is a false statement, the number is not a solution to the equation. To check to see if 1 is a solution to the equation + = x 3 4, we do the following. We use the symbol ?= to indicate that we are in the process of checking to see if we have a true statement. + = + = = x 3 4 13 4 4 4 ? Substitute 1 for x. True The same number is obtained on both sides of the equal sign, so 1 is the solution. For the equation + = x 3 4, the only solution is 1. Any other value of x would result in the check being a false statement.
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