Patterns and Relations

Transcription

1 Where a, b and care integers. a(x.b)2c b:c,a.io ax bc :b,a:o ax=b the farm: P.2: Model and solve problems, concretely, pictorially and symbolically, using linear equations of P.1: Graph and analyze two-variable linear relations. Patterns and Relations

2

3 language. The letters you are seeing are NOT some new bqn T PANIC those a little later. line on a graph, like in the number trick exercise. We will look at A linear relation is a relation whose plotted points lie in a straight for snacks. people (n) at $5 each with an additional $20 flat rate charge -as a way of representing an unknown number of example: 5n + 20 saying it s a number statement that includes a variable). operations to numbers and variables. (This is a fancy way of Algebraic expression: the result of applying arithmetic We see variables in algebraic expressions. quantity (number). The quantity (number) can vary or change. Variable: a letter that represents an unknown yet...they are called variables. The letters stand for a number we don t know Why Are There Letters in my Numbers??

4 variables. Table of values: a table displaying the relationship between two graph. Our first step is to make a table of values from a linear equation. Before we can begin graphing, we need to have something to put on a by plugging in a value for x, we get the relational value for y. expression 2x + 5 is a relational expression, and when we solve it variable x, so it is relational to (dependent on) x. The Relational value: the value of y depends on the unknown for x to find out the relational value y. We could continue this chart or substitute any unknown we want the same) 2x+5ory(theyare when we change the value of x, so we try different For this linear equation we need to know the value of y values of x to see what we get for y. variables into a linear equation and recording the result. We get this information by substituting actual numbers instead of Ex) y 2x + 5 Tables of Values

5 y:4x-3 y: 2x + 5 Create a Table of Values for each of the following relations: text p # 1-5 or 5-9 wkst. 9.1 yx-3

6 - - C Date: a Table of Values GOAL Create a table of values for a given linear relation Complete the table of values for each equation. a) b) 2. A rectangle is three times wider than it is long. a) Write an equation that you can use to determine the length when you know the width. b) Write an equation that you can use to determine the perimeter when you know the width. c) Complete the table of values for the perimeters of rectangles with the given widths. d) Widihof rectangle (cm) Perimeter of rectangle (cm) Determine the perimeter of a rectangle with a width of 45 cm. Explain your steps. At-Home You can represent a linear situation using an equation and a table of values. For example, lunch for up to 24 people costs $2 for each person, plus $4. Write an equation to represent this situation. Let n represent the number of people and c represent the total cost. The equãfiówisc=2n+4. Complete the table of values. Substitute values for n into the equation to determine the values for c.the value of n cannot be greater than 24. Number of Caoff people, n lunch, c ($) ci 76 Chapter 9: Linear Relations and Linear Equations Copyright 2009 by Nelson Education Ltd.

7 variable that goes on the x-axis. Hint: the relational value that goes on the y-axis depends on the value is not called y. You need to know which is which. TIP: sometimes the variable is not called x and the relational x y 2x the relational value y on the y-axis. We plot the unknown value (variable) x on the x-axis; we plot graph them. list of coordinates, or values for x and y. Now it is time to So, now that we have finished our table of values, we have a Graphing

8 Cartesian Plane t I - For a linear relation, a straight line may be drawn to connect the coordinates. What does this tell us? 1. We could extend the graph to find the matching coordinate for any given coordinate. (Interpolate) 2.We could use a graph to write a rule for a given situation or to make a table of values. (Work backwards to solve problems). (Extrapolate)

9 Graphing from a Table of Values y4x-7 Graphofy4x-7 * I * ±r td : :i:a: ±-- ±, 1: - it C3n 5 : : :j: : : :i:i: :j :t. 0 wkst 9.2 text p # 1-5 or 5-9

10 Date: Linear Relations GOAL Construct a graph from the equation of a given linear relation. 1. a) Complete the table of values. 7 G+ 3m b) Graph the points from your table on a Cartesian coordinate system I f EH 1 i zr _-LL- --j----t- L I I I I I Q 1 At-Home You can make a graph of a linear equation using a table of values. For example, the table of values for c = 2n + 4 is shown. a Plot n on the horizontal axis and c on the vertical axis. The n-axis will go from 0 to at least 24. The c-axis will go from 0 to at least I II. 40 C 24 I., II. 16 j U h n Copyright by Nelson Education t.td. Chapter 9: Linear Relations and Linear Equations 77

11 Solving Equations What is the difference between an expression and an equation? An expression is a combination of numbers and mathematical operations. It may include a variable. Ex) 4n + 6 We evaluate expressions for different values of n (or whatever you are calling your unknown variable). An algebraic equation is simply a number sentence that includes a variable and an equal sign (). (equation: equals...get it?) Algebraic equations show a relation, or a rule that allows you to use one number to get information about another number. ex) T 5n + 20 Total cost (T) of party with an unknown number of people (n) at $5 each with an additional $20 flat rate charge for snacks.

12 Iyj Eli Algetiles We can use this to help us solve algebra questionsl QQç C) C) C) group of 5 ç : C) of 3 is the same as a t 5 together with a group j Agraupf2pu1 other side. Equals () means that everything on one side is the same as everything on the the equation we are solving. The key is to ensure we are keeping When solving equations we can use algebra tiles or diagrams to model balance. Solve Equations Using biagrams and Models to

13 -)-fles-ti-yb To model the question: 13 3x 4 First we choose the appropriate tiles to create our question LY*i Then we begin to isolate our variable (try to get it Eu

14 Try These! 14: - 6 I -x I I i L-Y I 14: 3x + 5 wkst 9.4 text p # 1-5 or 5-8

15 GOAL to Represent Copyright by Nelson Education Ltd. Chapter 9: Linear Relations and Linear Equations 79 c) d = n 10, when d = 0 b) y=2x+160,wheny=212 Left side = Right side I = 12 = 3( 4) a) y=3x,wheny=18 3(( 6)+2) 12 Left side: Right side: that you would use. diagram? Explain your choice(s) and show the diagrams Check: 3. Which of these equations would you solve by drawing a If d+ 2 = 4, then dmustbe 6. LJ \_J\/ must equal 4. 3( 4) = 12, so each group of equal groups of L...J Ri-- 1OO=-12 equation. Write the equation. Then continue the solution. 2. The diagram is the beginning of a solution to an represent the equation. t = 12. Draw a picture to solve t = 3(d + 2), when drawing a diagram. For example, t = 2(n 3), when t = 20 You can solve an equation by 1. Solve the equation using a diagram. At-Home ordered pair. Draw a diagram to determine the missing value in an

16 REMEMBER! An equation is an expression that has an answer, or, a mathematical statement that contains an equal sign. Both sides of the equation are equal to each other, like two sides of a balanced scale. Ex) 4n+6=14 We can also solve an equation by isolating the variable and finding what it is equal to. Ex) 4n subtract 6 from both sides 4n:: divide both sides by 4 n 2 we have solved it!!

17 Simple Rules To solve an equation we basically undo everything that was done to the variable. If we have added constants (4n + 6) we subtract that amount from both sides C. If we have subtracted or negative constants (4n - we add that number to both sides. 6) If the variable is divided (..L) we multiply both sides of the equation by the divisor. 3 If a variable is multiplied (2n) we divide both sides by the multiplier.

18 How do we solve (or undo) each of the following to find the value of variable. 254x 5 3x-3: wkst 9.6 text p # 1-2, 6, 9 or 8-12

19 Date: Equations Symbolically GOAL Solve a linear equation symbolically. 1. Solve each equation and record your steps. Verify each solution. a) 2x 1 = 11 Check b) 3(b 8) = 6 Check c) 4m + 7 = 51 Check At-Home )!I You can solve a linear equation, such as 3x + 4 = 10, symbolically. Isolate the unknown variable on one side of the equation by adding, subtracting, multiplying, or dividing. For example, 3x + 4 3x + 4 = 10 4 = x = 6 3x 6 3 _3 x= 2 Substitute to check your answer. 3(2) + 4 = = 10 d)= 5 Check 2. Solve 5(x + 8) = 50 in two different ways. Show what you did in each solution. Solution 1 Solution 2 Copyright by Nelson Education Ltd. Chapter 9: Linear Relations and Linear Equations 81

20 The bistributive Property So, you know how to solve regular two step equations that look like this: 5n-4: 22 And even ones that look like this: 2n/ But can you solve equations that look like this one? 3(n - 4): 24 Yes, you can using the distributive property of numbers. That is a property that allows you to remove the brackets by multiplying each of the terms inside the brackets by the term outside of the brackets. 3(n-4):24 (3 x n) + (3 x _4): 24 3n (-12): 24 Now it looks familiar! 3n (-12): n:36 3 n 12

21 Let s try one more: + n) z 30 0 n=2

22 1) 8p-2p 6) -4y-8y Simplifying Algebraic Expressions Teacher: Date: Math-Aids.Com 5) -3(4+5k) 10) 5-6z-8+4z 4) -6w+3(7-2w) 9) 7x+2-8x y+3y 8) 4-9b+7 2) -6(7r+5) 7) -3(-8 5p)-9 Name: Score:

23 Solving Problems with Algebra When solving problems, we can sometime use an algebraic equation to help us solve the problem. Ex. The senior admission to the Royal Tyrell Museum is $2 less than an adult admission. A youth admission is $2 more than half the senior admission. The youth admission is $6. What is the senior and adult admission? 1. Understand the problem. I know the cost of the youth admission. I can use that to figure out the senior admission and then use the senior admission to figure out the adult. 2. Make a plan. I will write an equation to find out the senior admission and then write an equation to find out the adult admission. 3. Carry out the plan. y youth a adult s senior S y Look back. boes your answers make sense? Is the senior $2 less than the adult? Is the youth $2 less than half the senior? If not, you need to go back and check your work. wkst. 9.8 text p. 408 all

24 GOAL cal Reasoning Copyright 2009 by Nelson Education ltd. Chapter 9: Linear Relations and Linear Equations 83 $200. How much doestrip A cost? 9 days.trip C is twice the cost of Trip B, minus $1750. Trip B is twice the cost of Trip A, minus between three options. Trip A lasts for 5 days. Trip B lasts for 7 days.trip C costs $2850 for 4. John and his family are planning a trip to Vancouver, British Columbia. They are choosing his bank account. Today, he has $42 in his bank account. surrr of their ages wilt be 42. Wow old is Kathy? 2. Kathy is twice as old as Lisa.Three years from now, the 3. Each week, Ben deposits the same amount of money into Check your answer. Solve the equation. the form of an equation. Expressthe information in 2. Make a Plan solve the problem. 1. Understand the Problem age. How old is Sally? steps will help you: logical reasoning, the following 1. Sally s father is 45. He is 15 years older than twice Sally s 3. Carry Out the Plan 4. Look Back When solving problems using Search the question for information necessary to Use logical reasoning to solve the following problems. At-Home IIi) Three weeks from now, he will have $210. How much money does Ben deposit each week? Solve problems that involve equations using logical reasoning. Problems Using Date: