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Advanced Math
Learning Objectives By the end of this lesson, you will be able to: •
Identify the point of intersection as a solution to a system of equations
and solve for that point of intersection algebraically •
Solve equations involving factorials
•
Calculate the possible number of combinations or permutations
•
Calculate dimensions, surface area, and volume of three-dimensional figures
Systems of Equations A system of equations is a set of two or more linear equations.
Advanced systems of equations could include quadratic
equations, inequalities, and other non-linear equations. A system of equations can be solved by graphing or by using algebra.
Systems of Equations The solution to a system of equations is where the two lines intersect.
A system of equations may have no solution, one solution, or
infinite solutions. No solution
One solution
Infinitely many solutions
Solve Using Substitution Solve the system of equations x + y = 9 and 2x – 3y = 8 algebraically.
Step 1: Get x by itself in either equation xy 9 x y 9
Step 2: Substitute the equivalent expression into the other equation to solve for y 2x 3y 8
2(y 9) 3y 8 2y 18 3y 8
Step 3: Substitute y into either equation to solve for x x2 9 x7
5y 18 8 5y 10 y2
(7, 2)
Solve Using Combination Solve the system of equations x + y = 9 and 2x – 3y = 8 algebraically.
Step 1: Make the coefficients of one variable equal xy 9
Step 2: Add or subtract the equations to eliminate one variable 2x 2y 18 - 2x 3y 8
2(x y 9) 2x 2y 18
5y 10 y2
Step 3: Substitute y back into an equation to solve for x x (2) 9
x7
(7, 2)
Mixed Practice 1. Where does the line with the equation x – 2y = 4 intersect with the line with the equation 6y + 5x = 4? A. (2, –1)
B. (3, –2) C. (–2, 1) D. (–1, 2)
Mixed Practice 2. Find the solution to the equations y = 3x – 15 and x + y = 13.
A. (–1, 12)
B. (7, 6) C. (7, 20) D. (14, –1)
Mixed Practice 3. Which of the following is an equation of a line that intersects with the line y = 4x + 2 ? A. 2y = 8x + 2
B. y = 4x – 2 C. y = –4x + 2 D.
1 2
y = 2x – 7
Factorials The factorial of an integer is what you get when you multiply that integer by all the positive integers smaller than it.
Example: The factorial of 3 is 3 × 2 × 1 = 6. The factorial operation is represented by an exclamation mark: !
Example: 5! = 5 × 4 × 3 × 2 × 120 1= Factorials are used when working with combinations, permutations, and the fundamental counting principle.
Your calculator can solve factorials, combinations, and permutations for you!
Counting Techniques in Math A combination represents the number of ways there are to pick a small group out of a big group. For a combination, the order does not matter. n!
The formula for combinations is
r ! n r !
.
A permutation is like a combination—it counts how many ways there are to pick a small group out of a big group—but in a permutation, objects are ordered or assigned to specific positions after being chosen, so order does matter.
The formula for permutations is
n!
n r !
.
Counting Techniques in Math You can use the TI-30XS MultiView to skip the formula as long as you understand what the “n” and the “r” stand for in the calculator.
Example: Vladimir must select 3 of his 6 Science Cup students to compete as a
team in the National Science Cup Championships. How many different teams could Vladimir select?
Science Cup teams = order does not matter n = number of total items = 6 r = number of items being chosen for each combination = 3 combination = 6 nCr 3 = 20
Fundamental Counting Principle The fundamental counting principle states that the total number of possible outcomes for two or more events, positions or choices can be found by multiplying the possible number of outcomes together.
This principle will help you solve most permutation questions. Look for the phrases “how many different” or “how many distinct” to know when to use the fundamental counting principle. Follow these simple steps to use the fundamental counting principle: Step 1: Draw a blank to represent each event, position or choice. Step 2: Fill in the number of possibilities for each. Step 3: Multiply.
Mixed Practice 4. A chef will make a soup with five ingredients: one of four meats, one of four vegetables, one of four kinds of noodles, one of four kinds of broth, and one of four spices. How many different combinations of ingredients could the chef put into the soup? A.
20
B.
256
C.
1,024
D.
3,125
Mixed Practice 5. Soraya has been given six tasks to do at work, but she has time to complete only four of them. She must decide in what order to do the tasks. How many possible orderings of four tasks are available to Soraya? A. 15 B. 24 C. 360 D. 720
Mixed Practice 6. A club with 24 members is forming a three-person committee to plan a fund-raiser. What is the number of different three-person committees that can be formed? A.
12144
B.
2024
C.
8
D.
6
Three-Dimensional Figures Three-dimensional (3D) figures are shapes that have height, width and depth. The measure of space inside the shape is called the volume, and the measure of the total area of the faces is called the surface area.
rectangular prism
sphere
right prism*
cube
pyramid
cylinder
cone
Volume and Surface Area For certain 3D figures, you can apply a formula to find the surface area and volume
These formulas are provided, but consider memorizing them:
Mixed Practice 7. A box has two identical rectangular bases. Find the surface area and volume of a closed box with the dimensions below.
Mixed Practice 8. A pyramid is a three-dimensional object with four triangle faces that connect to the same vertex. Find the surface area of this pyramid.
Mixed Practice 9. A pyramid is a three-dimensional object with four triangle faces that connect to the same vertex. Find the volume of this pyramid.
Mixed Practice 10. A cylinder has two circular bases connected by a curved surface. Find the surface area and volume of this cylinder.
Mixed Practice 11. A cone is similar to a cylinder. The curved side of a cone slants inward so that it meets at a point, or vertex. Find the surface area of the first cone and the volume of the second cone below.
Mixed Practice 12. A sphere is a round solid figure where every point on the surface is the same distance from the center. Find the surface area and volume.
Learning Objectives Now that you have completed this lesson, you should be able to: •
Identify the point of intersection as a solution to a system of equations
and solve for that point of intersection algebraically •
Solve equations involving factorials
•
Calculate the possible number of combinations or permutations
•
Calculate dimensions, surface area, and volume of three-dimensional figures
Questions? Effort and practice are the keys to Test Day Success. Now that you have completed this lesson, you should reinforce these topics by completing practice questions.
If you have questions between live sessions, email your GED experts at
[email protected].
Happy studying!
Answer Key 1. Where does the line with the equation x – 2y = 4 intersect with the line with the equation 6y + 5x = 4? A. (2, –1)
B. (3, –2)
x 2y 4 x 2y 4
6y 5x 4
6y 5(2y 4) 4
C. (–2, 1)
6y 10y 20 4
D. (–1, 2)
16y 20 4 16y 16 y 1
Answer Key 2. Find the solution to the equations y = 3x – 15 and x + y = 13.
x y 13
A. (–1, 12)
y x 13
B. (7, 6) C. (7, 20) D. (14, –1)
-
y 3x 15 y x 13
y 3x 15 y x 13
0 4x 28 4x 28 x7
(7) y 13 y6
Answer Key 3. Which of the following is an equation of a line that intersects with the line y = 4x + 2 ? A. 2y = 8x + 2
B. y = 4x – 2 C. y = –4x + 2 D.
1 2
y = 2x – 7
Recall that when equations are in y = mx + b form, ‘b’ represents the y-intercept. y = 4x + 2 y-intercept: (0, 2)
Answer Key 4. A chef will make a soup with five ingredients: one of four meats, one of four vegetables, one of four kinds of noodles, one of four kinds of broth, and one of four spices. How many different combinations of ingredients could the chef put into the soup?
“How many different” = use the fundamental counting principle!
A.
20
B.
256
C.
1,024
4 ×4 ×4 × 4 × 4
D.
3,125
meat veg noodle broth spice
= 45 = 1,024
Answer Key 5. Soraya has been given six tasks to do at work, but she has time to complete only four of them. She must decide in what order to do the tasks. How many possible orderings of four tasks are available to Soraya? A. 15
“Orderings of the four tasks” = tasks in a row, order does matter
B. 24
Use the permutation (nPr) button on your calculator!
C. 360
What is n? 6
D. 720
What is r? 4 Plug it in:
6 nPr 4 = 360
Answer Key 6. A club with 24 members is forming a three-person committee to plan a fund-raiser. What is the number of different three-person committees that can be formed? A.
12144
“Number of three-person committees” = groups of people, order doesn’t
B.
2024
matter
C.
8
Use the combination (nCr) button on your calculator!
D.
6
What is n? 24 What is r? 3 Plug it in:
24 nCr 3 = 2024
Answer Key 7. A box has two identical rectangular bases. Find the surface area and volume of a closed box with the dimensions below.
Surface area = combined area of all six sides = 2lw + 2lh + 2wh Volume = length × width × height = lwh SA = 2 (4) (5) + 2 (4) (3) + 2 (5) (3)
V = lwh
= 40 + 24 + 30
= (4) (5) (3)
= 94 square feet
= 60 cubic feet
Answer Key 8. A pyramid is a three-dimensional object with four triangle faces that connect to the same vertex. Find the surface area of this pyramid.
Surface area
1 2
perimeter of base slant height area of base SA
1 2
ps B
p = perimeter of square base = 3 + 3 + 3 + 3 = 12
s = slant
= 5 inches
B = area of square base =
32
= 9 square inches
1 2
ps B
SA
1 2
(12 )(5) 9
SA = 6 (5) + 9
SA = 30 + 9 SA = 39 in²
Answer Key 9. A pyramid is a three-dimensional object with four triangle faces that connect to the same vertex. Find the volume of this pyramid.
Volume
1 3
area of base height perpendicular to base
Volume V
1 3 1 3
Bh (5 5) 6
50 cubic cm
1 3
Bh
Answer Key 10. A cylinder has two circular bases connected by a curved surface. Find the surface area and volume of this cylinder.
Surface area = 2π × radius × height + 2π × (radius)2 = 2πrh + 2πr2 Volume = π × (radius)2 × height = πr2h SA = 2πrh + 2πr2
V = πr2h
= 2π (5) (3) + 2π (5)2
= π (5)2 × (3)
= 2π (15) + 2π (25)
= 75π cubic units
= 30π + 50π = 80π square units
Answer Key 11. A cone is similar to a cylinder. The curved side of a cone slants inward so that it meets at a point, or vertex. Find the surface area of the first cone and the volume of the second cone below.
Surface area = π × radius × slant height + π × (radius) 2 = πrs + πr2 Volume
1 3
(radius) 2 height perpendicular to base
SA = πrs + πr2 = π (8) (4) + π
V= (8)2
= 32π + 64π
= 96π square units
V=
1 3 1 3
1 3
r 2 h
πr2h π (22) (9)
V = 12π cubic inches
Answer Key 12. A sphere is a round solid figure where every point on the surface is the same distance from the center. Find the surface area and volume.
Surface area = 4π × (radius)2 = 4πr2 Volume
4 3
(radius) 3
4 3
r 3
SA = 4πr2 = 4π (10)2 = 400π square units
V= =
4 3
4 3
πr3 π (103)
≈ 1333.33π cubic units
