Use your book glossary to find the definitions EQUATION: LINEAR EQUATION: an equation with exponent of 1 (in 1 variable) (in 2 variables) A statement of equality between 2 quantities INDEPENDENT VARIABLE: 4. DEPENDENT VARIABLE:
IDENTITY: One side of equal sign is the same as the other. All values of the variable produce a true statement. CONDITIONAL: Equation that is true for only certain values of the variable.
For the following, decide whether the equation is an identity or conditional and write the answer on the line. a. x + 2 = 7 (can x be anything? Or just one thing?) b. x – 3 = y (can x be anything? Can y be anything? Or just one thing?) conditional conditional
c. ab = ba d. P = 2l + 2w e. 2(x + 4) = 2x + 8 f. 4r – 1r = 3r identity Which are equations in 1 variable? conditional identity Which are equations in 2 variables? identity
Translate the following into equations. Let x be the number. One fourth of a number is 24. A number multiplied by 3 and then added to 6 gets 15. The difference between fourteen and twice a number is negative 8. (1/4)x = 24 6 + 3x = 15 14 – 2x = -8
For the following, translate the algebraic equation into an English sentence. Can you think of another way to write the same thing? x – 2 = -4 9x = 72 2 less than a number is negative four. Nine times a number is 72.
For the following, define the variables and find the equation The cost of renting a car at $35 flat fee and $0.11 per mile. The total cost of an item bought in Michigan; including tax. x = number of miles y = cost or renting a car y = 35 + 0.11x x = ticket cost of an item y = total cost of item y = x + 0.06x
The next set of sentences contains a phrase that describes a need for grouping symbols. Underline the phrase and then write the equation. Five times the difference between nine and a number is –35. The four times the sum of a number and 5 is 24. c. The total cost of an air-compressor is $5 for the first day plus $30 for each additional day. 5(9 – x) = -35 4(x + 5) = 24 y = 5 + 30(x – 1) * x= number of days
Birthday Party Exploration Emma wants to have her birthday party at Valley Bowling Alley. Her mother can rent a party room, let the kids rent shoes and play 3 games for $20 plus $10 per child. Estimate the total cost if 5 of Emma’s friends come to her party. Estimate the total cost if 15 friends come to Emma’s party. (she must be a very popular girl!) Explain how you found the costs in #1 and 2.
Her mother can rent a party room, let the kids rent shoes and play 3 games for $20 plus $10 per child. Write an equation representing the total costs of Emma’s birthday party at Valley Bowling Alley. Let t = the total cost of the bowling alley party (excluding cake, decorations…) and let n = the number of Emma’s friends who attend the party. t = Estimate the number of friends Emma can invite for $240. You only need to estimate to answer this question. Which variable do you know in #5? Where would you put it, if you wanted to use the equation? Try to set up the equation.
Which variable do you know in #1. Estimate the total cost if 5 of Emma’s friends come to her party. Try to set up the equation for that situation. Estimate the total cost if 5 of Emma’s friends come to her party. Do you think these 2 equations are the same of different? Why?
3.2 Solving Linear With Algebraic Notation
When my daughter was in 3rd grade, she had a problem that said: What number is added to 7 to get 15. I told her, “Emma, think 15 – 7 = ? “15 – 7 = 8”, she said. “Does 7 + 8 = 15?” She thought for a minute, then “YES!”
What’s My Number ? When I add 8 to my number, I get 20. What’s my number? What should I tell my daughter to think? Can you make an equation out of that problem? When I divide my number by 2, I get 6. What’s my number? Can you make an equation out of that problem?
When solving linear equations, we work backward To move things across the equal sign to ultimately get the variable by itself. Reverse the order of operations. What is the last order of operations? What comes before that?
SOLUTION – a value for the variable that makes the equation true. 3x – 2 = 1 Check for x = 0, x = 1 2 – 6x = 2 Check for x = -1, x = 1, x = 0 SOLUTION SET– all solutions to an equation. SOLVING – a process where you use inverse operations.
ADDITION PROPERTY for EQUATIONS if a = b, then a + c = b + c. You can ADD (SUBTRACT)the same thing to both sides of an equation. MULTIPLICATION PROPERTY for EQUATIONS if a = b, then ac = bc. You can MULTIPLY (DIVIDE)the same thing to both sides of an equation.
x + 8 = 20 What do you think we need to do 1st ? x + 8 = 20 -8 -8 x + 0 = 12 x = 12
White beans are positive numbers Black beans are negative numbers ve numbers Candy is the variable White and Black beans cancel each other out. ### x - ##### = -### ##### ##### ### x = ## X = ##/###
MAKE A PLAN 4x = 2 Subtract 2 from both sides 1. The 2 is added to the variable term 5 = 2 – 6x x is multiplied by –6 divide both sides by -6 **NOTE: you can change the subtraction on the variable to addition
Look in your book: Page 133 #56 Page 143 #82, 83, 84
3.3 Solving Equations Tables, Graphs, Symbols
EQUIVALENT – when you do something to both sides it remains equivalent 3x + 5 = 7 is equivalent to 3x = 2 (subtracted 5 both sides) Tables and Graphs-- give us a numerical and visual picture of what solving an equation means.
Solving Equations With Tables and Graphs Worksheet
CALCULATOR STEPS Y1 = left side Y2 = right side Window (set it) Graph GRAPHS: See page 147 y = 3x - 2 9 = 3x - 2 y=9 4 = 3x - 2 y=4 1 = 3x - 2 y=1 y=-5 -5 = 3x - 2
GRAPHS: y = 4 - x 4 = 4 - x y=4 1 = 4 - x y=1 -7 = 4 - x y= -7 Go to pg 153 - #10 Book/Calc.
SYMBOLS: (page 154) (#18) 3(x– 1) = -3 Does this look like linear form yet? What should we do 1st to get to linear form? x = 0 (#26) x = 5 (#30) 6 – 5(x – 1) = 31 **canchange subtraction to adding a (-) x = -4
How does a table show the solution to the equation? Look at y column for what is known and over to x column for answer. How does a graph show the solution to the equation? Look at y axis, go across to line, then down to solution off x axis. Which of the 3 methods would you prefer for 5 + 9x = 21? Table or Symbols– not a nice round number for graph Which of the 3 methods would you prefer for 5(x – 7)2 + x = 20? Graph or Table – a lot of work for symbols.
3.4 Solving Linear Equations With Variables on BOTH sides
Balance Beans White beans are positive numbers Black beans are negative numbers Candy is the variable White and Black beans cancel each other out. 3x – 5 = -3 + x # # # x - # # # # # = - # # # + #x - #x___________________ - #x ## x - # # # # # = - # # # Try these in your group: 3x – 1 = x + 1 5x + 6 = 3x + 2 6x – 6 = 10 – 2x TRY: 2(x + 1) = 4x - 8 ## x - # # # # # = - # # # + # # # # # +# # # # # # # x = # # X = # #/# # = #
Solving Symbolically 6 – 5x = 3(1 – x) Always clear parentheses 1st Collect all variables on 1 side Add 5x to both sides-why? Collect all constants on the other side. Solve for x.
Calculator Techniques WorkSheet
APPLICATIONS Page 165 # 46
In New York City, a U-Haul 24-foot moving truck costs $39.95 plus 1.59 per mile for 6 hours. The same truck from Budget costs $69.99 plus $0.89 per mile. U-Haul Budget y = 39.95 + 1.59x y = 69.99 + 0.89x State the equation by setting the two cost equations equal to each other. Solve and explain the meaning of the solution. What does the output of the point of intersection mean?
Tables and Graphs Worksheet (in groups) Look at Warm-Up Page 166
3.5 Solving Linear Inequalities In 1 Variable
SYMBOLIC FOR THE FOLLOWING, DETERMINE IF THE INEQUALITY IS TRUE OR FALSE AFTER THE OPERATION IS DONE. 4 < 10 ADD 2 to both sides. True/False 4 < 10 ADD -2 to both sides True/False 4 < 10 MULTIPLY 2 to both sides. True/False 4 < 10 MULTIPLY -2 to both sides. True/False 4 < 10 DIVIDE 2 to both sides. True/False Try -2. True/False
GENERAL RULE for INEQUALITIES The inequality stays the same when adding and subtracting across the inequality sign. The inequality ONLY CHANGES when you Multiply or Divide both sides by a negative number.
TABLES Find where they are the same Look at the behavior of the numbers before and after that point x – 1 < 2 What set of values makes the inequality true? Let’s graph on a number line: 0 2 4 Which side of 2 makes a true statement? Test the points on either side Look at graphs
TABLES Find where they are the same Look at the behavior of the numbers before and after that point 5 < 2 – 3x What set of values makes the inequality true? Let’s graph on a number line: -3 -1 0 Which side of -1 makes a true statement? Test the points on either side Look at graphs Try: 3 – 3x > 2 – 2x
GRAPHS Find the point of intersection Check to see which graph has the larger y values BEFORE the intersection Look what happens to the y values of that line AFTER the intersection Which side of the intersection would make the inequality true?
APPLICATIONS The graph below represents the profit for Companies A and B if they sell x units of their product. Complete the following table by using the graph to estimate.
Which company has the larger profit when 20,000 units are produced? Which company has the larger profit when 45,000 units are produced? Approximate from the graph, when the two companies’ profits are equal. How many units must be sold for Company A’s profits to be larger than Company B’s profits?