Go to butenhoffmath.com and check out the algebra notes video: A 6-1 AStd3b: I can solve systems of equations using the ‘Equal Values Method’ Butenhoff’s account Butenhoff has 100$ to start with but spends 10$ a week. 1. Equation: 3. Create tables that show when the accounts have the same value. 4. Sketch a graph that shows when the accounts have the same value. 5. Use the ‘Equal Values Method’ to algebraically show when the two accounts have the same value. Jamieson’s account Jamieson has 0$ to start with but saves 10$ a week. 2. Equation:
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Go to butenhoffmath.com and check out the algebra notes video: A 6-1 AStd3b: I can solve systems of equations using the ‘Equal Values Method’ Butenhoff’s account Butenhoff has 100$ to start with but spends 10$ a week. 1. Equation:
3. Create tables that show when the accounts have the same value.
4. Sketch a graph that shows when the accounts have the same value.
5. Use the ‘Equal Values Method’ to algebraically show when the two accounts have the same value.
Jamieson’s account Jamieson has 0$ to start with but saves 10$ a week. 2. Equation:
A 6-1 Name ___________________________________BDFM?________Why?______________________________ 1. What is today’s learning target? Johnson’s account Johnson has 60$ to start with but spends 5$ a week. 2. Equation:
4. Create tables that show when the accounts have the same value.
5. Sketch a graph that shows when the accounts have the same value.
6. Use the ‘Equal Values Method’ to algebraically show when the two accounts have the same value.
Ordaz’s account Ordaz has 0$ to start with but saves 5$ a week. 3. Equation: 7. Find the POI 7−= xy and 53 +−= xy by… 8. Find the POI of 34 −= xy and 4+−= xy by… …graphing. POI ________
…graphing. POI ________
…using Algebra
…using Algebra.
Solution____________.
Solution____________.
9. Write one sentence that describes how you use the ‘Equal Values Method’ to find the point of intersection of two lines. Solve 10. Alejandro starts with 50$ and earns 125$ a week. Angelina starts with 1050$ but loses 75$ a week. When will they have the same amount of money? How much do they have? Alejandro’s equation__________________________. Angelina’s equation___________________________.
11. Desmond started with $20 and adds to his account (deposits) $10 every month. Jasmine started with $110, but only deposits $7 each month. How many months will it take before Desmond has saved as much as Jasmine? How much do they have? Desmond’s equation______________________________ Jasmine’s equation_______________________________
12. Find the POI of
€
y = 3x − 9 and
€
y = −x + 7… 13. Find the POI of
€
y = 3x − 5 and
€
y = −12x + 2 by…
…graphing. POI ________
…graphing. POI ________
…using Algebra
…using Algebra.
Solution____________.
Solution____________.
Go to butenhoffmath.com and check out the algebra notes video: A 6-2 AStd3b: I can solve two-variable equations 1. Solve for x:6x + 9 = 4x + 2
2. Solve for x: 2x +3y =12
3. Solve for y: 2x +3y =12
4. Solve for x: 5x + 5(2x +1) = 35
A 6-2 Name ___________________________________BDFM?________Why?______________________________ 1. What is today’s learning target? Solve for x. 2. 5x −3= 2x + 9 3. 2x + 5= −2x + 45 4. 2x + 5= −2x + 45 5. −10x −8 = −4x − 50
6. Write one sentence that describes how you solve for x when there are only x’s. Solve for x. 7. 5x + 2y =15 8. 3x − 2y = 24 9. 5x + 5y = 30 10. −4x + 4y = 24
11. Write one sentence that describes how you solve for x when there are both x’s and y’s. Solve for y. 12. 5x + 2y =15 13. 3x − 2y = 24 14. 5x + 5y = 30 15. −4x + 4y = 24
16. Write one sentence that describes how you solve for y when there are both x’s and y’s.
Solve for x. 17. 4(x + 2)+3x = 36
17. 5(2x + 5)+3x = 51 19. 2x +3(x +3) =19 20. 2x +3(4x −1) = 25
21. Find the POI of
€
y = 3x − 5 and
€
y = −12x + 2 by…
…graphing. POI ________
…using Algebra POI ________
22. Solve for y and then graph: 3x + 2y =12
Slope Y-int
Equation
23. Write one sentence to describe when it may be helpful to be able to solve for y.
Go to butenhoffmath.com and check out the algebra notes video: A 6-3 AStd3b: I can solve systems of equations using the ‘Substitution’ method 1. Find the POI.
€
y = 2x +15x + 5y = 35
This system is gross because
€
y = 2x +15x + 5y = 35
This system is nice because
€
y = 2x +1
5x + 5(y) = 35
Before substitution.
€
y = 2x +15x + 5(y) = 35
After substitution.
€
y = 2x +15x + 5(2x +1) = 35
Find the POI.
€
y = 2x +1
5x + 5y = 35
A 6-3 Name____________________________________BDFM?__________Why?____________________________ 1. What is today’s learning target? Substitute but do not solve .
2. y = 5x2y+3x =1
3. y = 67y+ 4x = 9
4.
€
y = 8x + 42x − 9y =1
5. 2x + y = 8x = 2y+1
6. x = 2y+14y− 9x = 7
7. Write one complete sentence that describes how you use ‘Substitution’ to find the point of intersection. Find the POI. Thx.
8. y = x + 24y+3x = 36
(4, 6)
9. y = 2x + 55y+3x = 51
(4, 9)
10. y = x +32x +3y =19
(2, 5)
Find the POI by graphing. 11. y = 2x −3 and y = −x + 9 12. y = 2x + 4 and y = 2x −3 13. y = 2x + 5 and y = 2x + 5
14. y = 4x −12x +3y = 25
(2, 7)
15. x = 2y+13x + y =17
(5, 2)
16. y = 3x +110x − 2y =18
(5, 16)
Solve. 17. Solve for x: 3x − 2y = 48
18. Solve for y: 3x − 2y = 48
19. Solve for x: −4x + 4y =12
20. Solve for y: −4x + 4y =12
Consider problem #16 from the front side of the paper. 21. What method did you use to find the POI on #16? Why?
22. Solve the second equation for y.
23. What method can you now use to find the POI? Why?
24. Find the POI.
25. Write one sentence to describe when it may be helpful to be able to solve for y.
Go to butenhoffmath.com and check out the algebra notes video: A 6-4 AStd3: I can solve systems of equations using ‘Elimination’ including multiplying one or both equations by a positive or negative number. Find the POI. Thx.
1. −4x + 2y = 24x +8y = 28
The x’s are eliminated because _________________.
2. x − y =14x + 2y =10
We multiply the first equation by
_______ to create _______.
3. 3x − 2y =105x +3y = 23
We multiply the first equation by
_______ to create _______.
We multiply the second equation by
_______ to create _______.
A 6-4 Name______________________________________BDFM?______Why?______________________________ AStd3: I can solve systems of equations using ‘Elimination’
1. 925123
=+
−=−
yxyx
2. 624
1134−=+−
=+
xyxy
3. Write one complete sentence describing how you use elimination to find the point of intersection of two lines. AStd3: I can solve systems of equations using ‘Elimination’ by multiplying one equation by a positive or negative number.
4. Consider 4x − 2y =12x + 6y =11
a. Multiply the first equation by 3.
b. Rewrite the system with your new equation over the second equation.
c. Eliminate and solve.
5. On #4a, why is multiplying by 3 a good choice? Is there another option? Find the POI.
6. 3x − 2y =102x + 4y =12
7. 4x −3y = 52x + y = 5
AStd3: I can solve systems of equations using ‘Elimination’ by multiplying both equations by a positive or negative number.
8. Consider 3x + 2y = 74x + 5y =14
a. Multiply the 1st equation by 5. b. Multiply the 2nd equation by -2.
c. Rewrite the system with your new equations.
d. Eliminate and solve.
9. Find the POI of 5x + 2y =172x −3y = 3
10. Jamieson has 20$ and is saving 6$ per week. Butenhoff has 150$ and is spending 4$ per week. When will they
have the same amount of money? How much money do they have?
11. You are part of a group of tourists visiting Portland who want to rent bikes to ride around the area. You arrive at
the bicycle rental shop and see that there are two different rental plans: Plan A: 15$ initial fee, plus 2$ per hour. Plan B: 5$ initial fee, plus 4$ per hour. Define two variables and write a system of equations to find out when are the costs of the two plans the same and what is the cost?
Go to butenhoffmath.com and check out the algebra notes video: A 6-5 AStd3b: I can solve systems of equations using ‘Elimination’ including multiplying one or both equations by a positive or negative number. AStd3a: I can model systems in multiple ways 1. Packs of ‘Glue Stics’ cost 10$ each. Jamieson buys 4 packs of glue sticks. Show the calculations for how much they cost. 2. Packs of ‘American Gold’ pencils cost 6$ each. Butenhoff buys 3 packs of pencils. Show the calculations for how much they cost. 3. How many packs did the teachers buy? How much did they spend overall?
4. Packs of ‘Glue Stics’ cost 10$ each, packs of ‘American Gold’ pencils cost 6$ each, and you spend 54$. You buy 7 packs overall. Define the variables. How many of each did you buy?
A 6-5 Name_________________________________BDFM?________Why?________________________________ AStd3b: I can solve systems of equations using ‘Elimination’ including multiplying one or both equations by a positive or negative number. AStd3a: I can model systems in multiple ways 1. Let D=the number of diamonds you buy. Let G=the number of gold nuggets you buy. Write the equation given the prices. DO NOT SOLVE. a. Diamonds are 12$. Gold nuggets are 23$. You spend 430$
b. Diamonds are 20$. Gold nuggets are 56$. You spend 1000$.
c. Altogether you buy 40 rocks.
d. Altogether you buy 121 rocks.
e. Diamonds are 100$. Gold nuggets 45$. You spend 1000$.
f. Altogether you 600 rocks.
g. Gold nuggets are 78$. Diamonds are 23$. You spend 345$.
h. Altogether you buy 10,000 rocks.
Solve. 2. Burritos cost 5$ each, tacos cost 2$ each, and you spend 59$. Altogether you buy 19 items. Define the variables. How many of each did you buy? 3. Cookies cost 3$ each, brownies cost 4$ and you spend 215$. Altogether you buy 60 items. Define the variables. How many of each did you buy?
4. Ordaz has 40$ and is saving 10$ per week. Johnson has 205$ and is spending 5$ per week. Define the
variables. When will they have the same amount of money? How much money do they have?
5. Find the point of intersection of y = −2x +8y = −2x +1"#$
by…
…graphing.
…equal values. What happens when you solve for x?
6. Find the point of intersection of y = 3x − 4y = 3x − 4"#$
by…
…graphing.
…equal values. What happens when you solve for x?
Find the point of intersection using substitution.
7. y = 4x +32y−8x = 6"#$
8. y = 3x +14y−12x = 20"#$
9. 3x = y− 26x = 4− 2y"#$
A 6-Review Name______________________________BDFM?________Why?_______________________________ 1. Write one complete sentence that describes what you learned this chapter. C-Level Find the ‘Point of Intersection (POI)’…
2. …of y = 4x −10y = −x + 5"#$
3. ...of y = x − 7y = −3x + 5"#$
4. …of y = 2x −32y+ 6x =14"#$
5. …of 4y+3x =11−4y+ 2x = −6"#$
B-Level
7. You are part of a group of tourists visiting Portland who want to rent bikes to ride around the area. You arrive at
the bicycle rental shop and see that there are two different rental plans: Plan A: 20$ initial fee, plus 1$ per hour. Plan B: 0$ initial fee, plus 5$ per hour. a. Define two variables and write a system of equations
b. Find out when are the costs of the two plans the same and what is the cost?
Find the POI.
8. y = 3x + 48x − 2y = 6"#$
9. 3x + 2y = 74x + 5y =14!"#
A-Level 10. Burritos cost 6$ each, tacos cost 3$ each, and you spend 63$. Altogether you buy 13 items. Define the variables. How many of each did you buy? Find the POI…
11. …of 4x −3y = 54x + 2y =10
12. …of 4x −3y = 54x + 2y =10
using a
different method than #11.
13. Refer back to #11 and #12. Which method do you think is more ‘efficient?’
14. Find the POI of 3x = y− 26x = 2y−10"#$
using any method.
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