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I CAN . . . ~solve systems of equations by graphing. lesson 5.1
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vocabulary { New Word Visual Clue Definition system of equations
system of equations solution of a system coincident more than one equation x + y = 9 x – y = -5 More than one equation Point that is true for all equations in the system Two lines that have the same slope and same y-intercept
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Graphing systems Step 1: Equations should be in slope-intercept form. ______________ Rewrite equation if needed. Step 2: ________ both lines. Step 3: Find the coordinates of the point where the two lines ____________. (touch) Graph intersect
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Example #1 (0, 3) 3 Solution: _________ or x = ___, y = ___
Find the coordinates of the point where the lines touch. (0, 3) 3 Solution: _________ or x = ___, y = ___
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Example #2 2 –1 (2, –1) Solution: ________ or x = ___, y = ____
Find the coordinates of the point where the lines touch. (2, –1) 2 –1 Solution: ________ or x = ___, y = ____
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Example #3 Parallel Solution: _______________ No solution
Find the coordinates of the point where the lines touch. Solution: _______________ No solution
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Example #4 They overlap! 2y = –4x +14 Solution: _________________
–4x –4x 2y = –4x +14 They overlap! The lines are coincident (identical) Solution: _________________ Infinitely many
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Your turn! One a) b) Parallel lines = No solution c)
Same line = Coincident = Infinitely many Example 1-1b
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Draw a picture for each type of solution.
Are you a master? Draw a picture for each type of solution. No solution One Solution Infinitely Many
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vocabulary { New Word Visual Clue Definition solve
solve 3x + 2 = 14 Solve for x x = 4 To find a value for each variable
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I CAN . . . ~solve systems of equations by substitution. lesson 5.2
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substitution Step 1: ___________ to get _______ equation with _____ variable. Step 2: _______ the equation for the first variable. Step 3: ____________ to find the second variable. Substitute one one Solve Substitute
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Example #1 5 5 2x + y = 15 2x + y = 15 (y) 2x + (3x) = 15 5x = 15
1. Substitute to get one equation with one variable 2x + (3x) = 15 5x = 15 2. Solve the equation for the 1st variable x = 3 3. Substitute to find the second variable 2x + y = 15 2(3) + y = 15
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Example #1 2(3) + y = 15 6 + y = 15 –6 –6 y = 9
3. Substitute to find the second variable 2(3) + y = 15 6 + y = 15 – –6 y = 9 Check your answer! Does 9 = 3(3)? Yes!
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Example #2 2x – 2 = x + 2 +2 +2 2x = x + 4 –x –x x = 4
1. One equation with one variable 2x – 2 = x + 2 2. Solve 2x = x + 4 –x –x x = 4
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Example #2 Given: y = x + 2 x y = 4 + 2 y = 6 x = 4
3. Substitute & solve for the other x = 4 y = x + 2 x y = 4 + 2 y = 6 Check your answer! Does 6 = 4 + 2? Yes!
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Example #3 Solution: (–1, 5) 5x + 2(3x + 8) = 5 y = 3x + 8
–16 –16 11x = –11 y = 5 x = –1 Solution: (–1, 5)
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Example #4 y = –2x + 7 Solution: (0, 7)
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Example #5 Solution:
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Example #6 Monday’s homework and Tuesday’s homework have a total of 24 problems. There are three times as many problems Tuesday as on Monday. How many are in each set? Write two equations that fit this situation. M + T = 24 T = 3M Solution: Monday had 6 problems & Tuesday had 18.
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Are you a master? What do you look for in a system of equations that would lend itself to using substitution instead of graphing?
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Warm-up Use substitution to solve 3x + y = 6 -2x + 6y = 4
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lesson 5.3 Learning Target :I CAN . . .
~solve systems of equations with elimination. lesson 5.3
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Elimination of a variable
Step 1: Start with x and y on the same side (standard form). _______________ Step 2: Turn it into an __________ problem. Step 3: Add ____________. One variable should cancel out. Step 4: _______ to find remaining variable. Step 5: _____________ to find the other variable. Ax + By = C addition equations Solve Substitute
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Example #1 x + y = 50 x – y = 16 y – y x + y = 50 33 + y = 50 –33 –33
The sum of two numbers is The difference of the numbers is 16. What are the numbers? x + y = 50 x – y = 16 y x + y = 50 – y 33 + y = 50 2x = 66 – –33 y = 17 x = 33 Solution: (33, 17)
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Example #2 –3x –3x + 4y = 12 3x –3x + 4(–15) = 12 –2y = 30 –2 –2
– –2 –3x – 60 = 12 y = –15 –3x = 72 x = –24 Solution: (–24, –15)
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Example #3 –3y 2x + 3y = 6 3y 2(3) + 3y = 6 6x = 18 6 + 3y = 6 6 6
– –6 x = 3 3y = 0 y = 0 Solution: (3, 0)
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Example #4 Some friends go to the movies. They purchase2 popcorns and 3 sodas and pay $ The next weekend they go again and purchase 5 popcorns and 3 sodas for $ How much does the popcorn cost? 2p + 3s = 25.50 5p + 3s = 43.50
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Are you a master? Determine which method would be best for solving each system (graphing, substitution, elimination) and why? Do not solve.
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I CAN . . . ~solve systems of equations with multiplication and elimination. lesson 5.4
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Warm UP
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Example #1 2x – y = 6 3x + 4y = –2 8x – 4y = 24 4 11x = 22
3(2) + 4y = –2 x = 2 6 + 4y = –2 – –6 4y = –8 y = –2
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Example #2 2x + 3y = 6 2 –2x – 4y = –10 x + 2y = 5 –y = –4 2x + 3y = 6
– –1 2x + 3(4) = 6 2x + 12 = 6 y = 4 –12 –12 2x = –6 x = –3
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Example #3 3 3x + y = 4 – x + 5y = 34 –14y = –98 –14 –14 3x + y = 4
– –14 3x + y = 4 3x + 7 = 4 y = 7 –7 –7 3x = –3 x = –1
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(Warmup in notes: Example #4 pg 12 2x – 6y = 8 –2x + 6y = 7 0 = 15
x by 2 2x – 6y = 8 –2x + 6y = 7 0 = 15 No solutions What if we had a true statement?
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Example #5 2x – 5y = 8 3x + 2y = 31 4x – 10y = 16 2 15x + 10y = 155 5
3(9) + 2y = 31 x = 9 27 + 2y = 31 – –27 2y = 4 y = 2
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Example #4 pg10 Some friends go to the movies. They purchase2 popcorns and 3 sodas and pay $ The next weekend they go again and purchase 5 popcorns and 3 sodas for $ How much does the popcorn cost? 2p + 3s = 25.50 5p + 3s = 43.50
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Are you a master? Complete the multiplication required to set up the following system for elimination. You do NOT need to solve it.
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Warm-Up 2x + 2y = 8 x = 2 - y c – 4d = 1 2c – 8d = 2
Which solving technique would you choose for the following two problems? Explain IN WRITING (at least 3 sentences) WHY you chose that technique. 2x + 2y = 8 x = 2 - y c – 4d = 1 2c – 8d = 2
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lesson 5.5- the weird stuff
I CAN . . . ~solve systems of equations and figure out what the MEANING of their answer is! lesson 5.5- the weird stuff
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Example #1 (page 13) 2x + 2y = 8 x = 2 - y
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Example #2 c – 4d = 1 2c – 8d = 2
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Example #3 2b – 4c = 8 -3b + 6c = -12
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Example #3-5
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Learning Target: Graph inequalities on the coordinate plane
Lesson 5.6 Learning Target: Graph inequalities on the coordinate plane
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Steps: 2. If < or > then shade If ≤ or ≥ then shade
1. Graph the given line – draw lightly 2. If < or > then shade If ≤ or ≥ then shade 3. Shading – make sure you have slope intercept form (y = mx + b) if ≤, < then shade below if ≥, > then shade above
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Example #1 y ≥ 4
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Example #2
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Example #3
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Example #4
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Are you a master? True or False.
I love solving systems of equations and inequalities.
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Tasks for today Holiday Elf Word Problems
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