real world systems of equations. purpose purpose use systems of equations to determine exact...
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Real World Systems of Equations
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Purpose
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Purpose• Use systems of equations to determine exact quantities that satisfy two requirements;
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Purpose• Use systems of equations to determine exact quantities that satisfy two requirements; hence the need for a system.
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Example (Don’t copy)Michelle has 12 total pets in her home that are either fish or birds.
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Example (Don’t copy)Michelle has 12 total pets in her home that are either fish or birds.- Are there multiple
possible solutions to this situation?
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AbsolutelyMichelle has 12 total pets in her home that are either fish or birds.- Are there multiple
possible solutions to this situation?
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AbsolutelyMichelle has 12 total pets in her home that are either fish or birds.
The # of fish is 3 less than twice the number of birds.
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Now we have one situation that
represents this…Michelle has 12 total pets in her home that are either fish or birds.
The # of fish is 3 less than twice the number of birds.
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Method to solve:
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Method to solve:1) Establish the relationships between the 2 quantities.
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Examples
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Examples
x + y = n
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Examples
• You will often be told the total number of objects involved in the problem. x + y = n
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Examples
• You will often be told the total number of objects involved in the problem. This is what we use to represent this, where n is the total number of objects.
x + y = n
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Examples
y = ax ± b
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Examples
• You will be told that the number of one object is “b more than a times” the other. y = ax ± b
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Examples
• You will be told that the number of one object is “b more than a times” the other. a can also be a fraction, and it can also be “b less than”.
y = ax ± b
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Examples
y = mx ±
b
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Examples
• m is some type of rate, while b is a starting point.
y = mx ±
b
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Examples
• m is some type of rate, while b is a starting point. In this type of problem, the solution is usually a break –even point or where two of these quantities are equal.
y = mx ±
b
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Examples
• When dealing with measures of 2 quantities, we often use this method, ax + by =
c
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Examples
• When dealing with measures of 2 quantities, we often use this method, where a and b represent the “weight” of each individual object and c is the “total”.
ax + by =
c
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Let’s return to our intro problem.
Michelle has 12 total pets in her home that are either fish or birds.
The # of fish is 3 less than twice the number of birds.
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Let’s create our equations:
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Let’s create our equations:
Michelle has 12 total pets in her home that are either fish or birds.
![Page 27: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/27.jpg)
Let’s create our equations:
x + y = 12Michelle has 12 total pets in her home that are either fish or birds.
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Let’s create our equations:
x + y = 12The # of fish is 3 less than twice the number of birds.
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Let’s create our equations:
x + y = 12
x = 2y – 3
The # of fish is 3 less than twice the number of birds.
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Let’s create our equations:
x + y = 12
x = 2y – 3
Now we solve using, most likely, what method?
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Let’s create our equations:
x + y = 12x = 2y – 3
Now we solve using, most likely, what method?
Substitution
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Let’s create our equations:
x + y = 12x = 2y – 3
(2y – 3) + y = 12
Now we solve using, most likely, what method?
Substitution
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Let’s create our equations:
x + y = 12x = 2y – 3
(2y – 3) + y = 12 3y – 3 = 12
Now we solve using, most likely, what method?
Substitution
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Let’s create our equations:
x + y = 12x = 2y – 3
(2y – 3) + y = 12 3y – 3 = 12 3y = 15
Now we solve using, most likely, what method?
Substitution
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Let’s create our equations:
x + y = 12x = 2y – 3
(2y – 3) + y = 12 3y – 3 = 12 3y = 15 y = 5
Now we solve using, most likely, what method?
Substitution
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Mental math should be used to find y:
x + y = 12x = 2y – 3
(2y – 3) + y = 12 3y – 3 = 12 3y = 15 y = 5
Now we solve using, most likely, what method?
Substitution
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Mental math should be used to find y:
x + y = 12x = 2y – 3
y = 5x = 7
Now we solve using, most likely, what method?
Substitution
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Common problem
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Common problem• Which one represents the
birds and which one represents the fish?
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Common problem• Which one represents the
birds and which one represents the fish?
5 birds
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Common problem• Which one represents the
birds and which one represents the fish?
5 birds7 fish
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As a result:
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As a result:• We often use variables that
abbreviate the categories of items we have.
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New example:
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New example:You are the manager of a shoe store. On Sunday morning, you are going over the sales receipts for the past week. They show that 240 pairs of walking shoes were sold. Style A sells for $66.95, and Style B sells for $84.95. The total receipts for the two types were $17,652. The cash register was supposed to keep track of the number of each type sold. It malfunctioned. Can you find out how many of each type were sold?
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Setting up the system:
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Setting up the system:What letters shall we use?
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Setting up the system:a and bWhat letters
shall we use?
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Setting up the system:a and bWhat letters
shall we use?
How many total shoes were sold?
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Setting up the system:a and b
240
What letters shall we use?
How many total shoes were sold?
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Setting up the system:a + b = 240
240
What letters shall we use?
How many total shoes were sold?
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Setting up the system:a + b = 240
240
What letters shall we use?
What does style A and style B cost?
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Setting up the system:a + b = 240
A: $66.95
What letters shall we use?
What does style A and style B cost?
![Page 54: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/54.jpg)
Setting up the system:a + b = 240
A: $66.95B: $84.95
What letters shall we use?
What does style A and style B cost?
![Page 55: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/55.jpg)
Setting up the system:a + b = 240
66.95a + 84.95b = 17652
A: $66.95B: $84.95
What letters shall we use?
What does style A and style B cost?
![Page 56: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/56.jpg)
If we want to use elimination:
a + b = 24066.95a + 84.95b =
17652
A: $66.95B: $84.95
What letters shall we use?
What does style A and style B cost?
![Page 57: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/57.jpg)
If we want to use elimination:
a + b = 24066.95a + 84.95b =
17652
A: $66.95B: $84.95
What types of variable terms do we need?
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If we want to use elimination:
a + b = 24066.95a + 84.95b =
17652
A: $66.95B: $84.95
What types of variable terms do we need?
Opposites
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If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
What types of variable terms do we need?
Opposites
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If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
Adding straight down:
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If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
18b = 1584
Adding straight down:
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If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
18b = 1584b = 88
![Page 63: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/63.jpg)
If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
18b = 1584b = 88
Subtracting from 240:
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If we want to use elimination:
-66.95a – 66.95b = -16068
66.95a + 84.95b = 17652
18b = 1584b = 88a = 152
Subtracting from 240:
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Let’s get out your books:
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Let’s get out your books:
• Page 398
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Breaking things into categories…
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Breaking things into categories…
You invest a total of $9000 in two funds paying 5% and 6% annual interest. The combined annual interest is $510. How much of the $9000 is invested in each fund?
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Model:
x + y = n
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Model:
f + s = 9000
x + y = n
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Model:
f + s = 9000
f represents amount invested in 5% and s represents amount invested in 6%.
x + y = n
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Model:
f + s = 9000.05f + .06s = 510
x + y = n
Ax + By =
C
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Model:
f + s = 9000.05f + .06s = 510
When dealing with percents, put the percent as a decimal in front of the variable.
x + y = n
Ax + By =
C
![Page 74: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/74.jpg)
Model:
f + s = 9000.05f + .06s = 510
Now a little substitution after a bit of solving for a variable.
x + y = n
Ax + By =
C
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Model:
f = 9000 – s .05f + .06s = 510
Now a little substitution after a bit of solving for a variable.
x + y = n
Ax + By =
C
![Page 76: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/76.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
Now a little substitution after a bit of solving for a variable.
x + y = n
Ax + By =
C
![Page 77: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/77.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
x + y = n
Ax + By =
C
![Page 78: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/78.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
s + 450 = 510
x + y = n
Ax + By =
C
![Page 79: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/79.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
.01s + 450 = 510.01s = 60
x + y = n
Ax + By =
C
![Page 80: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/80.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
.01s + 450 = 510.01s = 60s = 6000
x + y = n
Ax + By =
C
![Page 81: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/81.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
.01s + 450 = 510.01s = 60s = 6000
Mental
math does
the rest…
![Page 82: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/82.jpg)
Model:
f = 9000 – s .05(9000 – s) + .06s = 510
450 – 0.05s + .06s = 510
.01s + 450 = 510.01s = 60s = 6000f = 3000
Mental
math does
the rest…
![Page 83: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/83.jpg)
Thus, we conclude…
![Page 84: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/84.jpg)
Thus, we conclude…6000 was invested in the 6% accounts and 3000 was invested in the 5% accounts.
![Page 85: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/85.jpg)
Back to your books…
![Page 86: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/86.jpg)
Back to your books…Page 410
![Page 87: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/87.jpg)
Rate problemsp 391 in your book:
![Page 88: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/88.jpg)
Rate problemsp 391 in your book:
y = mx + b
![Page 89: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/89.jpg)
Rate problemsp 391 in your book:
y = 25x + 400
y = mx + b
![Page 90: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/90.jpg)
Rate problemsp 391 in your book:
y = 25x + 400
y = mx + by = 50x +
200
![Page 91: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/91.jpg)
Solved by graphing
1000
900
800
700
600
500
400
300
200 0 2 4 6 8 10 12
y = 25x +
400
y = 50x +
200
![Page 92: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/92.jpg)
Solved by graphing
1000
900
800
700
600
500
400
300
200 0 2 4 6 8 10 12
y = 25x +
400
y = 50x +
200
![Page 93: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/93.jpg)
Solved by graphing
1000
900
800
700
600
500
400
300
200 0 2 4 6 8 10 12
y = 25x +
400
y = 50x +
200
![Page 94: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/94.jpg)
Solved by graphing
1000
900
800
700
600
500
400
300
200 0 2 4 6 8 10 12
y = 25x +
400
y = 50x +
200
![Page 95: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/95.jpg)
Appears to be approx.
(8, 600)
1000
900
800
700
600
500
400
300
200 0 2 4 6 8 10 12
y = 25x +
400
y = 50x +
200
![Page 96: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/96.jpg)
Let’s verify algebraically…
![Page 97: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/97.jpg)
Let’s verify algebraically…
y = 25x + 400y = 50x + 200
![Page 98: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/98.jpg)
Let’s verify algebraically…
y = 25x + 400y = 50x + 200
25x + 400 = 50x + 200
![Page 99: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/99.jpg)
Let’s verify algebraically…
y = 25x + 400y = 50x + 200
25x + 400 = 50x + 200200 = 25x
![Page 100: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/100.jpg)
Let’s verify algebraically…
y = 25x + 400y = 50x + 200
25x + 400 = 50x + 200200 = 25x
8 = x
![Page 101: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/101.jpg)
Let’s verify algebraically…
y = 25(8) + 400y = 50x + 200
25x + 400 = 50x + 200200 = 25x
8 = x
![Page 102: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/102.jpg)
Let’s verify algebraically…
y = 25(8) + 400y = 600
25x + 400 = 50x + 200200 = 25x
8 = x
![Page 103: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/103.jpg)
Let’s verify algebraically…
y = 25(8) + 400y = 600
Meaning after 8 months, they both will have 600 current
visits, if they both continue at the same rate.
![Page 104: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/104.jpg)
How to interpret the solution to these…
![Page 105: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/105.jpg)
How to interpret the solution to these…
Comparing 2 rates, the solution is the point when both quantities will be the same & the amount both will be.
![Page 106: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/106.jpg)
How to interpret the solution to these…
Comparing 2 rates, the solution is the point when both quantities will be the same & the amount both will be.
When you’re talking business (comparing sale price with cost), this is the break-even point.
![Page 107: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/107.jpg)
P 393 # 25 (Further interpretation)
![Page 108: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/108.jpg)
P 393 # 25 (Further interpretation)
When you are asked which one is a “better deal”:
![Page 109: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/109.jpg)
P 393 # 25 (Further interpretation)
When you are asked which one is a “better deal”:
The one with the smaller b is the better buy before the solution.
![Page 110: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/110.jpg)
P 393 # 25 (Further interpretation)
When you are asked which one is a “better deal”:
The one with the smaller b is the better buy before the solution.
The one with the smaller m is the better buy after the solution.
![Page 111: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/111.jpg)
P 393 # 25 (Further interpretation)
In this case:
![Page 112: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/112.jpg)
P 393 # 25 (Further interpretation)
In this case:
Before 125,000 miles, Car A is the better buy;
![Page 113: Real World Systems of Equations. Purpose Purpose Use systems of equations to determine exact quantities that satisfy two requirements;](https://reader030.vdocuments.us/reader030/viewer/2022012918/56649d015503460f949d49f2/html5/thumbnails/113.jpg)
P 393 # 25 (Further interpretation)
In this case:
Before 125,000 miles, Car A is the better buy; after this point, Car B is the better buy.