This document contains examples of using variables in algebraic expressions and equations. It includes word problems involving variables representing unknown quantities like the number of hours worked, number of bus rides, cost of items, etc. Expressions are written to represent relationships between the variables and known values. Equations are set up and solved for the variable. Sample problems include calculating a tip amount, identifying properties of triangles, writing expressions for daily pay based on hours worked, and determining total costs based on item quantities and per unit prices.
Name _________________________ Score ______ ______1..docxlea6nklmattu
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
k = 1, (-1)^k (k + 11) = (-1)^(1) (1 + 11)= -1*(12) = -12
k = 2, (-1)^k (k + 11) = (-1)^(2) (2 + 11)= 1*(13) = 13
k = 3, (-1)^k (k + 11) = (-1)^(3) (3 + 11)= -1*(14) = -14
k = 4, (-1)^k (k + 11) = (-1)^(4) (4 + 11)= 1*(15) = 15
(-12)+(13)+(-14)+(15)=2
2.
Locate the foci of the ellipse. Show your work.
X^2=(x-h)^2, then h=0
Y^2=(x-k)^2, then k=0
The centre is (0,0)
X^2/36+y^2/11=1
When x=0 y^2/11=1; y=0
When y=0,x=0
X^2/36=1;x=0
11+c^2=36
C=5
Foci (5,0) and (-5,0)
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
x:
2y - x = 5
2y - 5 = x
so x = 2y - 5
-Plug this into 2nd equation:
(2y - 5)² + y² - 25 = 0
-Use FOIL to solve the (2y - 5)² part:
(2y - 5)(2y - 5)
4y² - 10y - 10y + 25
4y² - 20y + 25
So :
4y² - 20y + 25 + y² - 25 = 0
Which can be simplified to:
4y² + y² - 20y = 0
4y² + y² - 20y = 0
y(4y + y - 20) = 0
So, because of the 0 multiplication rule,
y=0
x= -5 (plug in y=0 to original equations:
2y - x = 5
2(0) - x = 5, so x= -5)
(-5,0)
Y(4y+y-20)=0
So, y=0 or 4y+y-20=0
5y-20=0
Y=4
X=2y-5 when y=4
X=8-5=3
(-5,0) (3,4)
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3,
f(x)
22
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
X=1/3, y=-(11/3),z=-(5/3)
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
7x+8y>=336
Short Answer Questions:
Type your answer below each question. Show your work.
8.
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
Show your work.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
S1=1(6*1^2-3(1)-1)/2=1
S2=1^2+4^2=17
S31^2+4^2+7^2=66
9.
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely. Show your work.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into .
Pre-Calculus Quarter 4 Exam
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
2.
Locate the foci of the ellipse. Show your work.
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3, f(x)
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8.
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 12 + 42 + 72 + . . . + (3n - 2)2 =
9.
A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?
12
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 2 + 5 + .
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
Name _________________________ Score ______ ______1..docxlea6nklmattu
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
k = 1, (-1)^k (k + 11) = (-1)^(1) (1 + 11)= -1*(12) = -12
k = 2, (-1)^k (k + 11) = (-1)^(2) (2 + 11)= 1*(13) = 13
k = 3, (-1)^k (k + 11) = (-1)^(3) (3 + 11)= -1*(14) = -14
k = 4, (-1)^k (k + 11) = (-1)^(4) (4 + 11)= 1*(15) = 15
(-12)+(13)+(-14)+(15)=2
2.
Locate the foci of the ellipse. Show your work.
X^2=(x-h)^2, then h=0
Y^2=(x-k)^2, then k=0
The centre is (0,0)
X^2/36+y^2/11=1
When x=0 y^2/11=1; y=0
When y=0,x=0
X^2/36=1;x=0
11+c^2=36
C=5
Foci (5,0) and (-5,0)
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
x:
2y - x = 5
2y - 5 = x
so x = 2y - 5
-Plug this into 2nd equation:
(2y - 5)² + y² - 25 = 0
-Use FOIL to solve the (2y - 5)² part:
(2y - 5)(2y - 5)
4y² - 10y - 10y + 25
4y² - 20y + 25
So :
4y² - 20y + 25 + y² - 25 = 0
Which can be simplified to:
4y² + y² - 20y = 0
4y² + y² - 20y = 0
y(4y + y - 20) = 0
So, because of the 0 multiplication rule,
y=0
x= -5 (plug in y=0 to original equations:
2y - x = 5
2(0) - x = 5, so x= -5)
(-5,0)
Y(4y+y-20)=0
So, y=0 or 4y+y-20=0
5y-20=0
Y=4
X=2y-5 when y=4
X=8-5=3
(-5,0) (3,4)
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3,
f(x)
22
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
X=1/3, y=-(11/3),z=-(5/3)
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
7x+8y>=336
Short Answer Questions:
Type your answer below each question. Show your work.
8.
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
Show your work.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
S1=1(6*1^2-3(1)-1)/2=1
S2=1^2+4^2=17
S31^2+4^2+7^2=66
9.
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely. Show your work.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into .
Pre-Calculus Quarter 4 Exam
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
2.
Locate the foci of the ellipse. Show your work.
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3, f(x)
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8.
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 12 + 42 + 72 + . . . + (3n - 2)2 =
9.
A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?
12
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 2 + 5 + .
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
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