Unit 0 Lessons 1-3 Evaluating expressions using order of operations By R. Portteus and By Miss Klien modified by LHope

Evaluating Expressions and Combining Like Terms

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.

How do you describe a variable expression? Variable Expression MeaningOperation 5x, 5·x, (5)(x) (same as x·5) 5 times xMultiplication 5 divided by x Division 5 + x (same as x + 5) 5 plus xAddition 5 – x5 minus xsubtraction

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 = = 7 b.n – 3 n – 3 = 4 – 3 = 1

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

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.

A A A A A A Now You Try… Evaluate each expression given that a = 6, b = 12, and c = ac 2.a ÷ c 3.a + b + c 4.ba 5.b – c 6.c ÷ b

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 = ac 4ac = 4(6)(3) = (24)(3) = 72

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

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 = = = 21

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

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

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

Expressions compared to Equations Expressions 8y, 16a/b, 4r + s, 7 Equations 3x + 4 = 6 3r = 9

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

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

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.

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.

Let’s try one… Simplify the expression. 4x + 5x x + 7 4x, 5x, and -2x -2 and 7 4x+5x-2x = 9x-2x = 7x -2+7 = 5 7x + 5

Another example… 10x – 4y + 3x 2 + 2x – 2y 3x 2 10x, + 2x -4y – 2y 3x x – 6y Remember you cannot combine terms with the same variable but different exponents.

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

You Try #1 Simplify the following: 1.5x + 3y - 6x + 4y + 3z 5x - 6x 3y + 4y 3z -x + 7y + 3z

You Try #2 Simplify the following: 2.3b - 3a - 5c + 4b 3b + 4b -3a -5c -3a + 7b – 5c

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 ab 2 3ab + ab 2 + 9

You Try #4 Simplify the following: 4.5xy – 2yx + 7y + 3x – 4xy + 2x 5xy - 2yx - 4xy 7y 3x + 2x -xy + 7y + 5x

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

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

Try 2x + 6 – x + 6y -5y x y 3(3x + 7) 9x (c – 3) + 2c 4c – c 6c -12 What is still confusing?

The Distributive Property You will be able to use the distributive property and simplify expressions with like terms

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.

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

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

x So we can say that 4(x+2) = 4x+8

Example using the distributive property

Another Example

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.

Try