MM150 Unit 3 Seminar Sections

2 3.1 Order of Operations 2

Definitions Algebra: a generalized form of arithmetic. Variables: letters used to represent numbers Constant: a number on it’s own or a symbol/letter that represents a fixed quantity. Algebraic expression: a collection of variables, numbers, parentheses, and/or operation symbols (+ or x). Expressions DO NOT have equal signs, “=“. Examples: Algebraic equation: is an algebraic expression that has an equal sign, “=“. Examples: 2 + x = 113y – 9 = 36 3

4 Evaluating Expressions Exponents: x y  , you can rewrite this as 2 7  x x x x = x 4  (2a)(2a)(2a) = (2a) 3  (x + 6)(x + 6) = (x + 6) 2 x^2 is the same as x 2 2^3 = 2 3 = 2*2*2 = 8 Be careful! (-2) 4 = (-2)(-2)(-2)(-2) = = -(2*2*2*2) = -16 (-2) 4 is not equal to

5 Order of Operations 1.Parentheses: Perform all operations inside parentheses or other grouping symbols (use the rules below). 2.Exponents: perform all exponential operations or find any roots. 3.Multiplication/Division: perform all multiplication or division whichever comes first from left to right. 4.Addition/Subtraction: perform all addition or subtraction whichever comes first from left to right. P lease E xcuse M y D ear A unt S ally PEMDAS 5 Perform the one that comes first from left to right

Example of Evaluating an Expression Evaluate the expression x 2 + 4x + 5 when x = 3. Solution:x 2 + 4x + 5 = (3) + 5 = = 26 Be sure to follow the Order of Operations! 6

Example of Substituting for Two Variables Evaluate when x = 3 and y = 4. Solution: Be sure to follow the Order of Operations! 7

8 Examples of Checking Solutions A.Determine if 9 is the solution to 2 + x = 11. We can check by substituting 9 for x. 2 + x = = = 11 B.Determine if 10 a solution to 3y - 9 = 36. We can check by substituting 10 for y. 3y - 9 = 36 3(10) - 9 = = =/= 36 8 This is a true statement, therefore 9 is a solution to 2 + x = 11 This is a false statement, therefore 10 is NOT a solution to 3y – 9 = 36

9 EVERYONE: page 111 #42 Cost of a Tour: The cost, in dollars, for Crescent City Tours to provide a tour for x people can be determined by the expression x. Determine the cost for Crescent City Tours to provide a tour for 75 people. 9 Cost = x Substitute Cost = (75) Multiply Cost = Add Cost = $426.25

Linear Equations in One Variable 10

Terms Terms: parts that are added or subtracted in an algebraic expression Terms can be:  Constants: 3, - 5, 0, ⅜,   Variables: a, b, c, x, y, z  Products: 3x, ab 2, - 99ay 5 Expressions can have:  one term (monomial): x5t - 10y  two terms (binomial): y s - 11  three terms (trinomial): x 2 + 7x - 10  four terms or more (polynomial): x 2 y + xy - 11y + 23 NOTE: Decreasing power of the variable. Coefficient: numerical part of the term. Example: in the term 3x, 3 is the coefficient Example: in - 99ay 5, - 99 is the coefficient 11

Like and Unlike Terms Like Terms: have the same variables with same exponents on the variables.  5x and 3x are like terms  6ab and -9ab are like terms  16x 2 and x 2 are like terms  -0.35ac 5 and -400ac 5 are like terms Unlike Terms: have different variables or different exponents on the variables.  5x and 3 are unlike terms  6b and -9c are unlike terms  16x and x 2 are unlike terms  -0.35a 5 c and -400ac 5 are unlike terms 12

Example: Combine Like Terms 8x + 4x = (8 + 4)x = 12x 5y - 6y = (5 - 6)y = -y x x + 9 = (1- 5)x + (15 + 9) = -4x x y x = (3 + 7)x + 6y + (2 - 4) = 10x + 6y - 2 Add or subtract the coefficients of like terms and KEEP the same variable part 13

14 EVERYONE: page 113 #32 Combine like terms: 6(r - 3) - 2(r + 5) (r - 3) - 2(r + 5) + 10Distribute = 6r r Combine like terms = 4r – 18Finished! Note that 4r and - 18 are unlike terms, therefore you cannot combine them

15 Addition Property of Equality For real numbers a, b, and c if a = b, then a + c = b + c. 15 Example: Solve x - 9 = 15 x - 9 = 15 x = addition property x + 0 = 24 x = 24

16 Subtraction Property of Equality For real numbers a, b, and c if a = b, then a - c = b – c. Example: Solve x + 11 = 19 x + 11 = 19 x = subtraction property x = 8

17 Multiplication Property of Equality For real numbers a, b, and c, where c =/= 0 if a = b, then a c = b c Example : Solve Multiplication Property 17

18 Division Property of Equality For real numbers a, b, and c, where c =/= 0 if a = b, then Example: Solve 4x = Division Property

19 Steps for Solving Equations 1. Simplify (clean up) both sides of the equation by: a.) Get rid of any fractions by multiplying both sides of the equation by the LCD. b.) Use the distributive property to get rid of parentheses when necessary. c.) Combine like terms on same side of equal sign when possible.  Equation will be in the form ax + b = cx + d 2. Collect all the variables on one side of the equal sign and all constants to the other side by using the addition/subtraction property.  Equation will be in the form ax = b 3. Solve for the variable using the division/multiplication property.  The resulting form will be x = c 19

EVERYONE: Solve for x 3x - 4 = 17 20

EVERYONE: Solve for x 21 = 6 + 3(x + 2) 21

EVERYONE: Solve for x 8x + 3 = 6x

Proportions A proportion is a statement of equality between two ratios. Cross Multiplication If then ad = bc, b =/= 0, d =/= 0. b cb c a da d 23

To Solve Application Problems Using Proportions 1.Represent the unknown quantity by a variable. 2.Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right-hand side of the equal sign. When setting up the proportion, the same respective quantities should occupy the same respective positions on the left and right. For example, an acceptable proportion might be 24

To Solve Application Problems Using Proportions (continued) 3.Once the proportion is properly written, drop the units and use cross multiplication to solve the equation. 4.Answer the question or questions asked using appropriate units. 25

Example A 50 pound bag of fertilizer will cover an area of 15,000 ft 2. How many pounds are needed to cover an area of 226,000 ft 2 ? 754 pounds of fertilizer would be needed. 26

Formulas 27

Perimeter The formula for the perimeter of a rectangle is Perimeter = 2(length) + 2(width) or P = 2L + 2W Use the formula to find the perimeter of a yard when L = 150 feet and W = 100 feet. P = 2L + 2W P = 2(150) + 2(100) P = P = 500 feet 28

Volume of a Cylinder The formula for the volume of a cylinder is V = (pi)(r 2 )(h). Use the formula to find the height of a cylinder with a radius of 6 inches and a volume of in 3. The height of the cylinder is 5 inches. 29

30 -9x + 4y = 11 9x - 9x + 4y = 9x y = 9x EVERYONE: Solve the equation for y -9x + 4y = 11

31 5x + 3y - 2z = 22 -5x + 5x + 3y - 2z = -5x y - 2z = -5x y - 2z + 2z = -5x z 3y = -5x + 2z EVERYONE: Solve the equation for y 5x + 3y - 2z = 22

EVERYONE: Solve the equation for y 3x + 8y - 9 = 0 32

Solve for b 2. 33

Applications of Linear Equations in One Variable 34

35 Translating to Math Six more than a number6 + x A number increased by 3x + 3 Four less than a numberx – 4 Twice a number2x Four times a number4x 3 decreased by a number3 – x The difference between a number and 5x – 5 Four less than 3 times a number3x – 4 Ten more than twice a number2x + 10 The sum of 5 times a number and 35x + 3 Eight times a number, decreased by 78x – 7 Six more than a number is 10x + 6 = 10 Five less than a number is 20x – 5 = 20 Twice a number, decreased by 6 is 122x – 6 = 12 A number decreased by 13 is 6 times the numberx – 13 = 6x 35

To Solve a Word Problem 1.Read the problem carefully at least twice to be sure that you understand it. 2.If possible, draw a sketch to help visualize the problem. 3.Determine which quantity you are being asked to find. Choose a letter to represent this unknown quantity. Write down exactly what this letter represents. 4.Write the word problem as an equation. 5.Solve the equation for the unknown quantity. 6.Answer the question or questions asked. 7.Check the solution.

Example The bill (parts and labor) for the repairs of a car was $ The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed? Let h = the number of hours billed Cost of parts + labor = total amount h =

Example continued The car was worked on for 3.5 hours. 38

Example Sandra Cone wants to fence in a rectangular region in her backyard for her lambs. She only has 184 feet of fencing to use for the perimeter of the region. What should the dimensions of the region be if she wants the length to be 8 feet greater than the width? 39

Example continued Example continued 184 feet of fencing, length 8 feet longer than width Let x = width of region Let x + 8 = length P = 2L + 2W x + 8 x The width of the region is 42 feet and the length is 50 feet. 40

41 Page 139 #34 PetSmart has a sale offering 10% off of all pet supplies. If Amanda spent $15.72 on pet supplies before tax, what was the price of the pet supplies before the discount? the price before discount will be called “x”. x - x (0.10) = x x = x = x ≈ The price is ≈ $17.47

42 Page 140 #46 A bookcase with three shelves is built by a student. If the height of the bookcase is to be 2 ft longer than the length of a shelf and the total amount of wood to be used is 32 ft, find the dimensions of the bookcase. Let x = width (length of shelf) and let x + 2 = height From picture in book, there are 4 pieces of wood for width and 2 pieces of wood for the height. 4(width) + 2(height) = total amount of wood 4x + 2(x + 2) = 32 4x + 2x + 4 = 32 6x + 4 = 32 6x = 28 x = 42 So, width of bookcase is ft and height is ft