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Slide 2 Unit 1 Lessons 1-3 Evaluating expressions using order of operations By R. Portteus and By Miss Klien modified by LHope Slide 3 Evaluating Expressions and Combining Like Terms Slide 4 Evaluating Expressions Vocabulary: Variable A symbol, usually a letter of the alphabet, such as the letter n, that is used to represent a number. Variable expression (A.K.A. - Algebraic Expression) An expression, such as n 5, that consists of one or more numbers and variables along with one or more arithmetic operations. (Note: No equal sign) Evaluate a Variable Expression write the expression, substitute a number for each variable, and simplify the result. Slide 5 How do you describe a variable expression? Variable Expression MeaningOperation 5x, 5x, (5)(x) (same as x5) 5 times xMultiplication 5 divided by x Division 5 + x (same as x + 5) 5 plus xAddition 5 x5 minus xsubtraction Slide 6 Substitute 4 for n. Simplify Simplify (means to solve the problem or perform as many of the indicated operations as possible.) Solution: Substitute 4 for n. Simplify Solution: Evaluate a Variable Expression Example 1: Evaluate each expression when n = 4. a. n + 3 n + 3 = 4 + 3 = 7 b.n 3 n 3 = 4 3 = 1 Slide 7 Substitute 8 for x. Simplify Solution: Using parenthesis is the preferred method to show multiplication. Additional ways to show multiplication are 5 8 and 5 x 8. Substitute 8 for x. Simplify Recall that division problems are also fractions this problem could be written as: Evaluate an Algebraic Expression Example 2: Evaluate each expression if x = 8. a.5x 5x = 5(8) = 40 b.x 4 x 4 = 8 4 = 2 Slide 8 Substitute 4 for x; 6 for y. simplify Solution: Evaluating More Expressions Example 3: Evaluate each expression if x = 4, y = 6, and z = 24. a.5xy 5xy = 5(4)(6) = 120 b. = 4 Solution: Substitute 24 for z; 6 for y. Simplify. Slide 9 A A A A A A Now You Try Evaluate each expression given that a = 6, b = 12, and c = 3. 1.4ac 2.a c 3.a + b + c 4.ba 5.b c 6.c b Slide 10 Substitute the value for a = 6 and c = 3 into the problem and multiply Click to return to You try it slide You Try #1 Evaluate each expression given that a = 6, b = 12, and c = 3. 1.4ac 4ac = 4(6)(3) = (24)(3) = 72 Slide 11 Substitute the value for a = 6 and c = 3 into the problem and divide Click to return to You try it slide You Try #2 Evaluate each expression given that a = 6, b = 12, and c = 3. 2.a c a c = 6 3 = 2 Slide 12 Substitute the value for a = 6, b=12, and c = 3 into the problem, then add. Click to return to You try it slide You Try #3 Evaluate each expression given that a = 6, b = 12, and c = 3. 3.a + b + c a + b + c = 6 + 12 + 3 = 18 + 3 = 21 Slide 13 Substitute the value for b=12 and a = 6 into the problem, then multiply. Click to return to You try it slide You Try #4 Evaluate each expression given that a = 6, b = 12, and c = 3. 4.ba ba = (12)(6) = 72 Slide 14 Substitute the value for b=12 and a = 3 into the problem, then subtract. Click to return to You try it slide You Try #5 Evaluate each expression given that a = 6, b = 12, and c = 3. 5.b - c b c = 12 3 = 9 Slide 15 Substitute the value for c=3 and b = 12 into the problem, then Divide Note: It is better to rewrite this division problem as a fraction. This fraction can now be reduced to its simplest form. Substitute the value for c=3 and b = 12 into the problem, then Divide Note: It is better to rewrite this division problem as a fraction. This fraction can now be reduced to its simplest form. Divide both numerator and denominator by the GCF = (3) to reduce this fraction. It is OK to have a fraction as an answer. Click to return to You try it slide You Try #6 Evaluate each expression given that a = 6, b = 12, and c = 3. 6.c b Slide 16 Expressions compared to Equations Expressions 8y, 16a/b, 4r + s, 7 Equations 3x + 4 = 6 3r = 9 Slide 17 Combining Like Terms Now that we have seen some algebraic expressions, we need to know how to simplify them. Vocabulary Like terms: In an expression, like terms are the terms that have the same variables, raised to the same powers (same exponents). i.e. 4x and -3x or 2y 2 and y 2 Coefficient: A constant that multiplies a variable. i.e. the 3 in 3a or the -1 in b Slide 18 Like Terms In a term that is the product of a number and a variable, the number is the coefficient of the variable. -1 is the coefficient of x 3 is the coefficient of Slide 19 Combining Like Terms In algebra we often get very long expressions, which we need to make simpler. Simpler expressions are easier to solve! To simplify an expression we collect like terms. Like terms include letters that are the same and numbers. Slide 20 Like Terms Like terms are terms in an expression that have the same variable raised to the same power. In the expression above, 5x and 3x are like terms, but 5x and x 2 are not like terms. The constant terms 4 and2 are also like terms. Slide 21 Lets try one Simplify the expression. 4x + 5x -2 - 2x + 7 4x, 5x, and -2x -2 and 7 4x+5x-2x = 9x-2x = 7x -2+7 = 5 7x + 5 Slide 22 Another example 10x 4y + 3x 2 + 2x 2y 3x 2 10x, + 2x -4y 2y 3x 2 + 12x 6y Remember you cannot combine terms with the same variable but different exponents. Slide 23 Now you try Simplify the following: 5x + 3y - 6x + 4y + 3z 3b - 3a - 5c + 4b 4ab 2a 2 b + 5 ab + ab 2 + 2a 2 b + 4 5xy 2yx + 7y + 3x 4xy + 2x A A A A Slide 24 You Try #1 Simplify the following: 1.5x + 3y - 6x + 4y + 3z 5x - 6x 3y + 4y 3z -x + 7y + 3z Slide 25 You Try #2 Simplify the following: 2.3b - 3a - 5c + 4b 3b + 4b -3a -5c -3a + 7b 5c Slide 26 You Try #3 Simplify the following: 3.4ab 2a 2 b + 5 ab + ab 2 + 2a 2 b + 4 4ab - ab -2a 2 b + 2a 2 b 5 + 4 ab 2 3ab + ab 2 + 9 Slide 27 You Try #4 Simplify the following: 4.5xy 2yx + 7y + 3x 4xy + 2x 5xy - 2yx - 4xy 7y 3x + 2x -xy + 7y + 5x Slide 28 Conclusion A variable or algebraic expression is an expression that consists of one or more ________ and _________ along with one or more ________ _________. (Note: No _______ sign) To evaluate an expression write the _________, substitute a _______ for each variable, and _________ the result. numbers variables arithmetic operationsequal expression number simplify Slide 29 Conclusion Continued In an expression, __________ are the terms that have the same ______ __, raised to the same ____ (same exponents). A coefficient is a number that ________ a variable. like terms variables power multiplies Slide 30 Try 2x + 6 x + 6y -5y x + 6 + y 3(3x + 7) 9x + 21 4(c 3) + 2c 4c 12 + 2c 6c -12 What is still confusing? Slide 31 The Distributive Property You will be able to use the distributive property and simplify expressions with like terms Slide 32 The Distributive Property The product of a and ( b+c): a(b+c) = ab + ac ex: 5(x + 2) = 5x + 10 (b + c)a = ba + ca ex: (x + 4)8 = 8x + 32 The product of a and (b-c): a(b-c) = ab ac ex: 4(x 7)= 4x 28 (b-c)a = ba ca ex: (x-5)9 = 9x - 45 Sharing what is Outside the parentheses with EVERYTHING INSIDE the parentheses. Slide 33 What You Already Know You have been using this property in a simplified form since third grade. Now we give it the algebraic term, and we extend it a bit. OR Slide 34 A Visual Example of the Distributive property Find the area of this rectangle. We could say that this is 4(x + 2) x +2 Or.. x2 4 Slide 35 x 2 4 4 So we can say that 4(x+2) = 4x+8 Slide 36 Example using the distributive property Slide 37 Another Example Slide 38 Slide 39 Slide 40 Like terms, continued The distributive property allows you to combine like terms that have variables by adding coefficients. An expression is simplified if it has no grouping symbols and if all the like terms have been combined. Slide 41 Try Slide 42 Rational Vs Irrational See Live Binders