CHAPTER 5 EQUATIONS FOR PROBLEM SOLVING Advanced Math

CHAPTER 5: EQUATIONS FOR PROBLEM SOLVING Advanced Math

Section 5. 1: Modeling Problem Situations Mathematical Models – graphs, tables, functions, equations, or inequalities that describe a situation Modeling – using mathematical models

Section 5. 1: Modeling Problem Situations (con’t) There are 5 ways to model a situation (equation). 1. 2. 3. 4. 5. Using a Table Using a Spreadsheet Using a Graph Using an Equation Using a Graphing Calculator **We will be using tables, graphs, and equations to model equations in class**

Sample 1 Model and solve this situation: A CD player costs $195, including tax. You already have $37 and can save $9 a week. After how many weeks can you buy the player?

Sample 1(con’t) Number of Weeks Total Saved ($) = 37 + 9 x (No. of wks) 0 37 1 37 + 9(1) = 46 2 37 + 9(2) = 55 3 37 + 9(3) = 64 * * 16 37 + 9(16) = 181 17 37 + 9(17) = 190 18 37 + 9(18) = 199

Sample 1 (con’t) The problem asks for the number of weeks until you will have $195. Let w = the number of weeks Then, 9 w = the amount in dollars you can save in w weeks, and 37 + 9 w = the total amount you will have saved in w weeks. You will have enough money to buy the compact disc player when the total amount saved equals the amount you need to purchase the CD player.

Sample 1 (con’t) 37 + 9 w = 195 -37 9 w = 158 9 9 w = 17. 6 18 weeks

Try this one on your own… Model and Solve this situation: Suppose you are reading a novel that has 378 pages. You are now on page 62. Starting tomorrow, you plan to read 20 pages a day. How many days will you need to finish.

Number of Days Total # of pages 62 + 20 x (# of days) 0 62 1 62 + 20(1) = 82 2 62 + 20(2) = 104 3 62 + 20(3) = 122 * * 15 62 + 20(15) = 362 16 62 + 20(16) = 382 17 62 + 20(17) = 402 62 + 20 D = 378 - 62 -62 20 D = 316 20 20 D = 15. 8 16 Days

Sample 2 The same model can be used to describe different situations. Describe a situation that could be modeled by the equation 5 = 3 + 0. 25 X For the expression 0. 25 X, think of a situation that involves 25% or a quarter of something.

Sample 2 (con’t) I ran 3 miles. How many times (x) around a quarter-mile track must I run to complete a 5 mile run? I have sold three dollars worth of cookies at the neighborhood bake sale. How many 25 cent cookies (x) do I need to sell to bring my sales total up to five dollars?

Try this one on your own… Describe a situation that could be modeled by the equation… 2 X + 90 = 180 The acute angles of a right triangle are congruent. What is the measure of each acute angle?

Section 5. 2: Opposites and the Distributive Property Simplify - (a – 5) Step One: Rewrite problem using a 1 -1 (a – 5) Step Two: Distribute the -1 -1 (a) – (-1)(5) Step Three: Simplify -a – (-5) Step Four: Subtracting a Negative is the same as adding its opposite. -a + 5

Sample 2 Simplify 5 m – (3 m –n) 5 m – 1 (3 m-n) 5 m – 3 m -1(-n) 5 m – 3 m + n 2 m + n

Try these on your own… Simplify: - (c + 6) -c - 6 Simplify: 26 - 3(h + 5) -3 h + 11

Sample 3 … Multistep Equations The temperature at noon today was 6 degrees. At 9 p. m. , the temperature was -2 degrees. How many degrees did the temperature drop today? 6–x=-2 -6 -6 Let x = the number of degrees -1 x = -8 -x = (-2 -6) -x = -8 -1 -1 X=8

Sample 4 … Multistep Equations Solve the equation: 12 + 6 x – 8 x = 20 12 – 2 x = 20 OR 12 + (-2 x) = 20 12 -2 x = 20 -12 -2 x = 8 -2 -2 x = -4

Try these on your own… The sea-level elevation for the entrance to a cave was -50 m. An exploration team reported 3 hours after entering the cave that they had descended to a level of -130 m. They did this in 2 equal stages. How many meters did they descend in each stage? -50 – 2 d = 130 D=40 Solve the equation: -7 x + 18 + 2 x = 33 X = -3

Sample 5 The sum of the measures of the angles of a convex polygon with N sides is 180(n-2). Can the sum of the measures of the angles of a convex polygon be 450 degrees? 180 (N-2) = 450 180 N – 360 = 450 180 N = 810 N = 4. 5 No, the sum of the measures of the angles of a convex polygon cannot be 450 degrees.

Section 5. 3: Variables on Both Sides Variable Terms – Terms of an expression that contain a variable

Solving Equations with Variables on Both Sides Solve: 4 x = 2 x + 6 Step One: Get all the variables on one side 4 x = 2 x + 6 -2 x 2 x = 6 Step Two: Solve 2 x = 6 2 2 x=3 Solve: 3 x = x + 16 -x -x 2 x = 16 2 2 X=8

Section 5. 4: Inequalities with One Variable When you multiply or divide by a negative number, the inequality sign flips. Open Circle: < or > Close Circle:

Sample 1 Solve and graph the inequality x + 2 < -1. Try this one on your own… n x + 2 < -1 -2 -2 x < -3 x – 5 > -1

Sample 2

Try these on your own…

Sample 3

Try this one on your own…

Sample 4: Modeling Situations with Inequalities Jamal can afford to spend $50 to buy some concert tickets and pay for parking. Model this situation. n n Tickets: $18. 00 Parking: $5. 00 Try this one on your own… n Katerina has $123 in her checking account. With the extra earnings from her job, she can deposit $50 a week. When she reaches $500 in her checking account, she plans to open a savings. Model this situation.

Sample 5: Modeling Situations with Inequalities Jamal borrowed $50 from his mom for the concert. He repays the loan at the rate of $6 per week. When will his debt be under $20? Try this one on your own. . n Yuri has 17 minutes of music on a 90 -minute cassette. How many 5 minute songs can he still get onto the cassette?

Section 5. 5: Rewriting Equations Solve D = rt for t. and Formulas Try this one on your own… n Solve A = 1/2 bh for b

Sample 2 Solve 2 x + y = 180 for y. Try this one on your own… n 5 f – 9 c = 160 for c

Sample 3 Solve P = 2 L + 2 W for W Try this one on your own… n W = 3 M – 4 K for K

Sample 4 Suppose there is an 8% sales tax on all items purchased at a craft supplies store. n Write a formula to show the total amount T, including tax, that you would pay for items at the store that cost C dollars altogether.

Try this one on your own… Suppose Felicia has saved $140 and plans to save an additional $15 each week out of the salary she makes at her parttime job. n Write a formula to show the total amount D that she has saved after W weeks.

Section 5. 6: Using Reciprocals Solve. n 15 = 6 h Try these on your own… n 14 w = 49 W = 3. 5 n (-2/3)x = 97 n (11/9)x = 165 X = 135 n 12 = (4/5)x + 4 n -10 = (-4/5)y + 6 Y = 20

Sample 2 Leela wants to use her new graphic calculator to see the graph of the equation 2 x + 3 y = 6. She pressed the (y=) key to enter the equation. Here is what she saw: : Y 1 = : Y 2 = : Y 3 = : Y 4 = Rewrite the equation so that Leela can enter it on her calculator.

Try this one on your own… Wong wants to graph the equation -3 x+4 y=12 on his graphing calculator. How can he rewrite the equation so that he can enter it. n y = 3 + (3/4)x

Roberto and two roommates ordered takeout shrimp, and the three agreed to split the cost (c) equally. Let s = Roberto’s share of the cost. n Write an equation that describes the situation.

Try this one on your own… Carlotte Mendez overhauled her tractor and mowed two of her five equal-sized fields. Overhauling her tractor took her 1. 5 hours. Let t = time it takes her to mow all five fields, and let s = the time for the work she has already done. n Write an equation that describes the situation. s = 1. 5 + (2/5)t

Section 5. 7: Area Formulas Parallelogram - Area = Base x Height Triangle – Area = ½ x Base x Height Trapezoid – Area = ½ x the sum of the bases x height Area = ½ x (b 1 + b 2) x height

Parallelogram Find the area of a parallelogram with base 10 cm and area of 25 cm squared. A = Bh 25 = 10 h 10 10 2. 5 = h

Trapezoid Step One: Draw a picture and label the parts Step Two: Fill in the area formula. A = ½(B 1 + B 2)h 15600 = ½(x + 2 x)130 15600 = ½ (3 x) 130 15600 = (1. 5 x)130 15600 = 195 x 195 80 = x Janelle Rose wants to buy a trapezoid plot of land. She knows that the border along the water is twice as long as the border along the street. The property is 130 feet tall and has a total area of 15600. How long is the border along the water?

Try these on your own… Find the area of the trapezoid whose has one base of 6 cm, another of 8 cm, and a height of 4. 3 cm. n The area is 30. 1 centimeters squared. The side face of the control tower at an airport are trapezoidal. Each side face has an area of 425 square feet. The edges along the floor each measure 18 feet. The trapezoids each have a height of 17 feet. What is the measure of each edge along the ceiling? n The edge along the ceiling each measure 32 feet.

Section 5. 8: Systems of Equations in Geometry The measure of an acute angle of a right triangle is four times the measure of the other acute angle. Find the measure of each angle. X + 4 X = 90 5 X=90 5 5 X = 18 Y = 4 X X + Y = 90 1 ST: 18 degrees 2 nd: 72 degrees

Try this one on your own… The measure of one acute angle of a right triangle is 12 degrees more than the measure of the other acute angle. Find the measure of each acute angle. Y = X + 12 X + Y = 90 X + (X + 12) = 90 X + 12 = 90 2 X + 12 = 90 -12 2 X = 78 2 2 X = 39 1 ST: 39 DEGREES 2 ND: 51 DEGREED

Solutions of Systems Two or more equations that state relationships that must all be true at the same time are called system of equations. Example: y = 4 x and x + y = 90 The values of the variable that make both equations true at the same time are the solution of a system. Example: x = 18; y = 72