Three points that are equivalent to the polar point (7,40°) are (7,400°), (7,-320°), and (7,760°). These points are equivalent because they all have the same magnitude of 7, but their angles differ by multiples of 360°.
In polar form, these points are written as:
- (7,400°) = 7cis(400°)
- (7,-320°) = 7cis(-320°)
- (7,760°) = 7cis(760°)
These points are equivalent because the angle in polar coordinates is measured modulo 360°. This means that any angle that differs by a multiple of 360° will be equivalent. For example, 40° is equivalent to 400° because 400° - 40° = 360°. Similarly, -320° is equivalent to 40° because -320° + 360° = 40°, and 760° is equivalent to 40° because 760° - 720° = 40°.
In conclusion, the three points that are equivalent to the polar point (7,40°) are (7,400°), (7,-320°), and (7,760°), and they are written in polar form as 7cis(400°), 7cis(-320°), and 7cis(760°).
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1.Solve 3^2x−1=27, showing steps 2 Solve 3^x∧2−3x=81, showing work
3 Solve 4^x+1=64, showing steps as 4 Solve 4^x+1=1/64, showing work as
1. To solve 3^(2x-1)=27, we first need to simplify 27 to power of 3: 27=3^3
3^(2x-1)=3^3
Next, we can use the property that if the bases are the same, the exponents must be equal:
2x-1=3
Solving for x, we get:
2x=4
x=2
2. To solve 3^(x^2-3x)=81, we first need to simplify 81 to the power of 3: 81=3^4
3^(x^2-3x)=3^4
Next, we can use the property that if the bases are the same, the exponents must be equal:
x^2-3x=4
Solving for x, we can use the quadratic formula:
x=(-b±√(b^2-4ac))/(2a)
x=(-(-3)±√((-3)^2-4(1)(-4)))/(2(1))
x=(3±√(9+16))/2
x=(3±5)/2
x=4 or x=-1
3. To solve 4^(x+1)=64, we first need to simplify 64 to the power of 4: 64=4^3
4^(x+1)=4^3
Next, we can use the property that if the bases are the same, the exponents must be equal:
x+1=3
Solving for x, we get:
x=2
4. To solve 4^(x+1)=1/64, we first need to simplify 1/64 to the power of 4: 1/64=4^(-3)
4^(x+1)=4^(-3)
Next, we can use the property that if the bases are the same, the exponents must be equal:
x+1=-3
Solving for x, we get:
x=-4
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2. Write an equation with the given characteristics: a. Goes through: \( (2,13) \) and \( (2,-11) \)
An equation passing through the points (2, 13) and (2, -11) is x = 2.
The equation of a line can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
To find the equation of the line that goes through the given points, we first need to find the slope. The slope can be calculated using the formula:
[tex]\[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\][/tex]
Plugging in the given points, we get:
[tex]\[m = \frac{-11 - 13}{2 - 2} = \frac{-24}{0}\][/tex]
Since the denominator is 0, the slope is undefined. This means that the line is vertical. A vertical line has the equation x = a, where a is the x-coordinate of any point on the line. Since both of the given points have the same x-coordinate of 2, the equation of the line is x = 2.
Therefore, the equation with the given characteristics is x = 2.
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olve the following linear programming problem. Restrict x ≥ 0 and y ≥ 0. Maximize f = 2x + 4y subject to x + y ≤ 7; 2x + y ≤ 12; y ≤ 4.
(x,y)=
f=
The solution to this linear programming problem is (x,y) = (3,4) and f = 22.
To solve this linear programming problem, we need to find the feasible region by graphing the constraints and then use the objective function to find the maximum value of f.
1. Graph the constraints:
x + y ≤ 7 can be rewritten as y ≤ -x + 7
2x + y ≤ 12 can be rewritten as y ≤ -2x + 12
y ≤ 4
2. Find the feasible region:
The feasible region is the area that satisfies all of the constraints. In this case, it is the area bounded by the three lines and the x and y axes.
3. Use the objective function to find the maximum value of f:
f = 2x + 4y
To find the maximum value of f, we need to find the corner points of the feasible region and plug them into the objective function. The corner points are (0,4), (3,4), and (4,3).
f(0,4) = 2(0) + 4(4) = 16
f(3,4) = 2(3) + 4(4) = 22
f(4,3) = 2(4) + 4(3) = 20
The maximum value of f is 22 at the point (3,4).
Therefore, the solution to this linear programming problem is (x,y) = (3,4) and f = 22.
Answer:
(x,y) = (3,4)
f = 22
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Can y’all please help a gurl out thanks
Answer:
The answer is B
Step-by-step explanation:
To solve this, we need to combine like terms.
(9c-8d) + (2c-6) + (-d+3)
9c + 2c - 8d - d - 6 + 3
11c - 8d - d - 6 + 3
11c - 9d - 6 + 3
11c - 9d - 3
help what is this 8y to the second power - 3y to the second power
Answer: 5y²
Step-by-step explanation:
Since 8y² and 3y² are similar terms, we can subtract the coefficients and keep the degree and variable the same.
8 - 3 = 5, so 8y² - 3y² = 5y²
Write down the value of the digit in colour as a fraction with a denominator that is a power of 10. a) 0,453 b)43,1 c)92,303 d)2,3214
Answer:
a) The digit in the hundredths place in 0.453 is 3. Therefore, the value of the digit in color is 3/100.
b) The digit in the tenths place in 43.1 is 1. Therefore, the value of the digit in color is 1/10.
c) The digit in the thousandths place in 92.303 is 3. Therefore, the value of the digit in color is 3/1000.
d) The digit in the ten-thousandths place in 2.3214 is 1. Therefore, the value of the digit in color is 1/10,000.
A toolbox is 2 ft high, and its width is 3 ft less than its length. If its volume is 80 ft³, find the length and width of the box.
The length and width of the toolbox are 8 feet and 5 feet respectively.
What is the length of the toolbox?The volume of a rectangular prism is expressed as;
V = w × h × l
Where w is the width, h is height and l is length.
Given that the volume of the toolbox is 80 cubic feet, so we can write:
V = w × h × l = 80ft³
Next, we know that the width is 3 feet less than the length, so we can write:
w = l - 3
Now we can substitute the second equation into the first equation to get an equation with just one variable:
V = w × h × l = l(l - 3)(2) = 80
Simplifying this equation, we get:
2l² - 6l - 80 = 0
We can solve this quadratic equation using the quadratic formula:
l = (-b ± √(b² - 4ac)) / 2a
where a = 2, b = -6, and c = -80. Plugging in these values, we get:
l = (6 ± √(6² - 4(2)(-80))) / 4
l = (6 ± √(676)) / 4
We take the positive value of l since the length must be positive, so we get:
l = (6 + 26) / 4
l = 8
Now we can use the second equation (w = l - 3) to find the width:
w = l - 3
w = 8 - 3
w = 5
Therefore, the length of the toolbox is 8 feet and the width is 5 feet.
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Classify the following variables.
discrete quantitative
quantitative continuous
qualitative nominal
qualitative ordinal
a. Number of newspapers sold in a day. b.Qualification of a newly elected politician (excellent, good, fair bad)
c. Gender of a student (male, female)
d. Number of students in a first-year classroom.
e. Number of a student (A000000000)
f. Olympic medal type (gold, silver, pronce)
The classification of the variables in the question above is as follows:
a. Number of newspapers sold in a day - discrete quantitative, as it is a countable number.b. Qualification of a newly elected politician (excellent, good, fair, bad) - qualitative ordinal, as it ranks or orders categories.c. Gender of a student (male, female) - qualitative nominal, as it is a categorical variable without a specific order.d. Number of students in a first-year classroom - discrete quantitative, as it is a countable number.e. Number of a student (A000000000) - qualitative nominal, as it is a categorical variable without a specific order.f. Olympic medal type (gold, silver, bronze) - qualitative ordinal, as it is a ranking or ordering of categories.Discrete quantitative data refers to data that can be counted in whole numbers and is often used to describe things that can be categorized into groups. like the number of people in a room or the number of cars in a parking lot.
Quantitative continuous data refers to data that can be measured on a continuous scale, such as time, temperature, or weight, like the temperature of a room or the weight of a person.
Qualitative nominal data refers to data that can be categorized into groups but cannot be measured or ranked, like the type of car someone drives or the color of someone's eyes.
Qualitative ordinal data refers to data that can be categorized into groups and can be ranked or ordered, like the level of education someone has achieved or the ranking of a sports team.
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Easton has a goal to complete 70% of the jumps on a skateboard course. If he has completed 18 out of 30 jumps already, how many more jumps does Easton need to complete to reach his goal?
Easton needs to complete 3 more jumps to reach his goal of completing 70% of the jumps on the skateboard course.
How to determine the additional number of jumps neededEaston's goal is to complete 70% of the jumps on the skateboard course. This means he needs to complete 70% of the total number of jumps.
We know that Easton has already completed 18 jumps out of 30, which is equivalent to completing 60% of the jumps.
The total number of jumps on the skateboard course is 30.
Easton's goal is to complete 70% of the jumps, which is:
0.7 x 30 = 21 jumps
Easton has already completed 18 jumps, so he needs to complete:
21 - 18 = 3 more jumps
So, he needs 3 more jumps
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(p ↔ r) → (∼q → (p ∧ r))
simply this by propostional algebra
answer must be verifiable by truth table
Truth Table
This statement can be simplified using proportional algebra as follows: (p ∨ ∼r) ∧ (q ∨ (p ∧ r)). This can be verified using a truth table.
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Using a scale of 1 inch : 16 feet, what are the blueprint dimensions of a building that is 70 feet × 90 feet?
0.27 inches X 0.2 inches
4.375 inches X 5.625 inches
1,120 inches X 1,440 inches
4 inches X 5 inches
The blueprint dimensions of a building that is 70 feet × 90 feet is: 4.375 inches X 5.625 inches.
What is scale ?Scale refers to the ratio or proportion between the dimensions of an object or a system in the real world and its representation in a model, drawing, or map. A scale is typically expressed as a ratio or a fraction, such as 1:100 or 1/4, which indicates the relationship between the size of the object in the real world and the size of its representation in the model or drawing.
According to given information :Using a scale of 1 inch : 16 feet means that every inch on the blueprint represents 16 feet in the actual building. To find the blueprint dimensions of a building that is 70 feet x 90 feet, we need to divide each dimension by 16.
The blueprint dimensions are:
70 feet ÷ 16 = 4.375 inches
90 feet ÷ 16 = 5.625 inches
Therefore, the blueprint dimensions of the building are approximately 4.375 inches by 5.625 inches.
The answer is: 4.375 inches X 5.625 inches.
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Math141 Spring 22 Do Home-Secling Di Uren Pit 01/01 12A Homework Section 7.2: Sampling Distributions Question 2.7.2.30 Part 2 of 2 1.3.6.7 and 9 ott 0.2.4.6, and even. Consider 24-tietoa random number table Compute porta and below HW Scon 1001N, 1.17 of 11 ports Points: 5 of 1 Save How many of the 24 digits would you expect to be even on average? 12. (Type an integer or a decimal. Do not round) b. If you actually courted, would you got exactly the number predicted in part ay? Explan OA Yes, because the sample is sufficiently large that the sample proportion will be the same as the population proportion OB. No, because samples will never have exactly the number predicted due to variation from sample to sample OC. No, beteuse while a sample might have exactly the number predicted, a sample could who have male or larger unbendus to variation from a OD, Yos, because samples will always match the population proportion
a) On average, 0.5 * 24 = 12 of the digits would be even. b) No, you would not necessarily get exactly the number predicted in part a. This is because there is always variation from sample to sample. While the expected value is 12
a. On average, you would expect 12 of the 24 digits to be even. This is because there are an equal number of even and odd digits (0, 2, 4, 6, 8 are even and 1, 3, 5, 7, 9 are odd), so the probability of a digit being even is 0.5. Therefore, on average, 0.5 * 24 = 12 of the digits would be even.
b. No, you would not necessarily get exactly the number predicted in part a. This is because there is always variation from sample to sample. While the expected value is 12, it is possible to get more or less than 12 even digits in a sample of 24.
This is due to the randomness of the random number table and the fact that samples do not always perfectly match the population proportion.
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HELP PLS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
-3/2
Step-by-step explanation:
To find:-
Slope of the line.Answer:-
We are interested in finding out the slope of the given line . We can see that the line passes through the points (1,0) and (-1,3) . For finding out the slope we can use the formula,
[tex]:\implies \sf m =\dfrac{y_2-y_1}{x_2-x_1} \\[/tex]
Now on substituting the respective values, we have;
[tex]:\implies \sf m = \dfrac{0-3}{1-(-1)} \\[/tex]
[tex]:\implies \sf m = \dfrac{-3}{1+1}\\[/tex]
[tex]:\implies \sf \pink{m=\dfrac{-3}{2}}\\[/tex]
Hence the slope of the line is -3/2.
A bag of mixed nuts contains almonds and hazelnuts. There are (6x+13) nuts in this particular bag, and (3x-7) of there are hazelnuts.
Which expression represents the number of almonds in the bag?
The expression that represents the number of almonds in the bag is (6x+13) - (3x-7)
What is Linear Equation?
A linear equation is a mathematical equation that represents a straight line on a graph. It is an equation in which the highest power of the variable is one. A general linear equation can be represented as y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.
Linear equations can be solved algebraically using techniques like substitution or elimination, and they can be used to model real-world situations involving linear relationships.
The expression that represents the number of almonds in the bag is (6x+13) - (3x-7) which simplifies to 3x + 20.
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Pls help!
In a video game each player earns 5 pts for reaching the next level and 15 pts for each coin collected. Make a table to show the relationship between the num of coins collected c and total pts p graph the ordered pairs and analyze the graph.
Answer:
Here is a table showing the relationship between the number of coins collected and the total points earned:
Number of coins (c) Total points (p)
0 0
1 20
2 35
3 50
4 65
5 80
6 95
7 110
8 125
9 140
10 155
To graph these ordered pairs, we can plot the number of coins collected (c) on the x-axis and the total points earned (p) on the y-axis. Then we can plot each ordered pair as a point on the graph and connect the points with a line. The resulting graph should show a linear relationship between c and p, with a slope of 15 and a y-intercept of 0.
Analyzing the graph, we can see that as the number of coins collected increases, the total points earned also increases at a constant rate. This suggests that collecting coins is an important part of the game, as it significantly increases the player's total score. Additionally, the slope of the line (15) tells us that each additional coin collected is worth 15 points, while reaching the next level is worth only 5 points.
help me below??? dont guess
Answer:
C is your answer
Step-by-step explanation:
WILL GIVE BRAINLIST TO BEST ANSWER
Given the recursive formula for a geometric sequence find the common ratio, the first five terms, the term named in the problem, and the explicit formula.
Show work
Find f(9)
11) f(n) = f(n - 1) x 4
f(1) = 4
Answer:
No solution
Step-by-step explanation:
Use the given functions to set up and simplify f(9)
Your friend asks you for help to check a geometry exercise. Below is vour friend's paper.
What error did your friend make? Explain.
The error is made because two or more shapes are said to be similar if they have common corresponding properties of sides and/ angles. The Option D is correct.
What is the explanation of the Geometric error?All similar shapes are two or more given shapes that share some properties in terms of their corresponding sides or angles. As a result, if the corresponding sides or angles are related, they are similar. It is important to note that similarity does not imply congruency.
When the sides and angles of the given triangles LMN and QRS are compared, it is clear that: angle Side Angle (ASA) similarity implies that the included side between two pairs of congruent angles is similar. As a result, the Option D explains the error in the diagram.
Missing options "a.Your friend's paper does not name the triangles correctly for them to be congruent. b. Your friend's paper shows that the SAS Postulate should be used to show congruence because a pair of congruent angles is included between two pairs of congruent sides. c. The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the triangles do not have two pairs of congruent angles. d. The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the included sides between the two pairs of congruent angles are not marked as congruent in both triangles.
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I need help on this asap!
The inequalities that represent the regular price of eyeglass frames will be 0.4r ≤ 120 and r ≥ 4(360/5).
How to calculate the InequalitiesIf we let r represent the regular price of the eyeglass frames, then we can write the following two inequalities based on the given information:
0.4r ≤ 120
This inequality represents the fact that the lowest regular price for the eyeglass frames is $120. We use 0.4 because a 60% discount means that the price paid is 40% of the regular price.
Also, r ≥ 4(360/5) as this inequality represents the fact that the highest regular price for the eyeglass frames is $360. We use 4(360/5) because a 75% discount means that the price paid is 25% of the regular price, which is equivalent to multiplying the regular price by 0.25. Then, 4 times that amount gives us the regular price, which is $360 in this case.
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For a given rational function, n < m. what does this mean about the graph of the function?
There are both horizontal and oblique asymptotes.
There is no horizontal asymptote.
There is a horizontal asymptote at y=a/b
There is a horizontal asymptote at y=0 .
For rational function, n < m. There is a horizontal asymptote at y = 0.
Explain about the rational function?A rational function is one whose denominator is not zero and which can be expressed as the ratio between two polynomials.
f(x) = p(x) / q(x)
A horizontal line is only an asymptote to the graph's extreme left and right. Anything beyond the vertical asymptotes as well as x-intercepts is considered to be "far" left or "far" right. The midpoint of horizontal asymptotes is not asymptotic. A horizontal asymptote can be crossed in the middle.By examining the degrees of a numerator (n) and denominator, we can locate the horizontal asymptote (m).
The horizontal asymptote is at y=0 if nm, the x-axis, is true.Y=an/bm is the horizontal asymptote if n=m. The ratio of the main coefficients, in other words.There isn't a horizontal asymptote if n>m. It has an oblique or slant asymptote, though, if n=m+1.Thus, For rational function, n < m. There is a horizontal asymptote at y = 0.
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Unit rates.
Calista ran 6.75 miles in 45 minutes. At this rate, his far can she run per minute?
Answer:0.15 miles a minute
Step-by-step explanation:
Her pace is 6 minutes and 40 seconds a mile, she is going 0.15 miles per minute and will go 9 miles an hour
Solve the equation to: x/4 - 16 = (-32)
Thaink you all who answered this!!
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PLEAAAASEEEE HELPP!!!!!!!!!!! I WILL GIVE BRAINLIEST!!!!!!!!!!
The median of class A is equal to 40.
The median of class B is equal to 8.
The difference between the median of the two classes is 32.
What is a dot plot?In Mathematics, a dot plot can be defined as a type of line plot that is typically used for the graphical representation of a data set above a number line, especially through the use of dots.
Based on the dot plot, the median of class A can be determined by sorting the data set in ascending order;
10, 10, 10, 20, 20, 30, 40, 40, 40, 40, 50, 50, 50, 70, 70
Median of class A = 40.
Based on the dot plot, the median of class B can be determined by sorting the data set in ascending order;
4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 12, 12, 16
Median of class B = 8.
For the difference, we have:
Difference = 40 - 8
Difference = 32.
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For ab+ay-b^(2)-by, (a) Factor out the GCF from the polynomial and identify the category in which the remaining poly
The final factored form of the polynomial is b(a-b) and the category of the remaining polynomial is a binomial.
The first step in factoring out the GCF from the polynomial ab+ay-b^(2)-by is to identify the greatest common factor of all the terms. In this case, the GCF is b. Once we have identified the GCF, we can factor it out from each term by dividing each term by the GCF. This gives us:
ab+ay-b^(2)-by = b(a+y)-b(b+y) = b(a+y-b-y)
Next, we can simplify the remaining polynomial by combining like terms:
b(a+y-b-y) = b(a-b)
Finally, we can identify the category of the remaining polynomial. Since it has two terms and each term has a degree of 1, the remaining polynomial is a binomial.
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Suppose that 30% of the applicants for a certain industrial job possess advanced training in
computer programming. Applicants are interviewed sequentially and are selected at random from
the pool. A) Find the probability that the first applicant with advanced training in programming is
found on the fifth interview. B) What is the expected number of applicants who need to be interviewed in order to find
the first one with advanced training
After calculating the probability we get, there is a 7.20 percent chance that the first candidate will be identified during the fifth interview and it should take 3.33 interviews to discover the first candidate.
The quantity X of repeated trials needed to obtain r successes with p probability in a binomial experiment is known as the negative binomial distribution.
When x succeeds after n tries, the probability is given by:
P(X-x) = Cn-1, x-1*p(power x) * (1 - p)power (n-x)
The number of unique combinations of x items from a set of n elements, Cn,x, is determined by the formula below.
Cn,x = n! / x! ( n! - x! )
This problem involves that:
Let's say that 30 percent of those who apply for a certain industrial position have advanced expertise in computer programming. That follows that.
(a) We need to determine the probability that the first applicant with advanced programming training will be discovered during the fifth interview.
This is the likelihood that it will take 5 attempts to get 1 success.
So, n=5, x=1
P(X-x) = Cn-1, x-1*p(power x) * (1 - p)power (n-x)
P(X-5) = C 4,0* (0.30)(power 1) * (0.70)power (4)
= 0.0720.
During the sixth interview, there is a 7.20% chance that the first candidate with advanced programming training will be discovered.
(b) To find how many applicants should be expected to be interviewed in order to select the first candidate with advanced training, calculate
It is stated by how many trials are anticipated to result in r success:
E = r/p
Thus with the value r=1
E = r/p
= 1/0.3
= 3.33.
For the first candidate with advanced training, it will take an average of 3.33 interviews.
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Solve the system of equations and choose the correct answer from the list of options. d + e = 1 −d + e = −5 Label the ordered pair as (d, e). (4 points) a (0, 0) b (3, −2) c (−2, 3) d (−3, 0)
Answer:
b (3, −2)
Step-by-step explanation:
You want the solution to the system of equations ...
d +e = 1-d +e = -5EliminationWe see the coefficients of d are opposites, so we can add the equations together to eliminate the d variable:
(d +e) +(-d +e) = (1) +(-5)
2e = -4 . . . . . . . . simplify
e = -2
Substituting into the first equation, we have ...
d +(-2) = 1
d = 3 . . . . . . add 2
The solution is (d, e) = (3, -2).
__
Additional comment
We also note that the coefficients of 'e' are identical, so we can eliminate 'e' as a variable by subtracting one equation from the other. We choose to subtract the equation with the lower value coefficient of 'd'.
(d +e) -(-d +e) = (1) -(-5)
2d = 6 . . . . . . . simplify
d = 3 . . . . . . . . divide by 2; matches the solution above.
The attached graph uses x and y, because those are the built-in independent and dependent variables. Your graphing calculator may let you define the variables as you wish.
Shea runs a carpet cleaning business. The average cost to shea per cleaning is $30. Shea charges $60 per cleaning. Shea’s fixed plus variable costs per month total $1,500. How many carpet cleaning does shea need to do per year to break even?
Shea needs to do 600 carpet cleaning per year to break even.
What is revenue?
Total revenue is the total amount of money earned by a business from the sale of its products or services during a particular period of time.
To break even, the total revenue Shea generates from the carpet cleaning business must be equal to the total cost of running the business.
Let's first calculate the total cost of running the business per year:
Total Cost = Fixed Costs + Variable Costs
Since the fixed plus variable costs per month total $1,500, the total cost per year would be:
Total Cost = $1,500 x 12
Total Cost = $18,000
Now, let's calculate the profit that Shea makes per cleaning:
Profit per Cleaning = Price per Cleaning - Cost per Cleaning
Profit per Cleaning = $60 - $30
Profit per Cleaning = $30
So, Shea makes a profit of $30 per cleaning.
To break even, the total profit generated by the number of cleanings Shea does per year should be equal to the total cost of running the business per year:
Total Profit = Total Revenue - Total Cost
If x is the number of carpet cleaning Shea needs to do per year to break even, then:
Total Revenue = Price per Cleaning x Number of Cleanings per Year = $60x
Total Profit = Profit per Cleaning x Number of Cleanings per Year = $30x
Setting Total Profit equal to Total Cost:
$30x = $18,000
x = $18,000 / $30
x = 600
Therefore, Shea needs to do 600 carpet cleaning per year to break even.
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Let A= (2 1)
(6 4)
(a) ExpressA−1as a product of elementary matrices. (b) ExpressAas a product of elementary matrices.
E1 x E2 x E4 = (2 0) (0 1) (1 0) (3 1)
(a) A-1 can be expressed as a product of elementary matrices by following these steps:
1. Create a matrix A' = A (2 0)
(0 1)
2. Create an elementary matrix E1 = (1 -2)
(0 1)
3. Multiply A' and E1 to get matrix E2 = (2 -4)
(0 1)
4. Create an elementary matrix E3 = (1 0)
(-3 1)
5. Multiply E2 and E3 to get matrix E4 = (2 0)
(-6 1)
6. Create an elementary matrix E5 = (1 0)
(0 2)
7. Multiply E4 and E5 to get A-1 = (2 0)
(-3 2)
Therefore, A-1 can be expressed as a product of elementary matrices, E1 x E3 x E5 = (2 0) (-6 1) (1 0) (0 2).
(b) A can be expressed as a product of elementary matrices by following these steps:
1. Create an elementary matrix E1 = (2 0)
(0 1)
2. Create an elementary matrix E2 = (1 1)
(0 1)
3. Multiply E1 and E2 to get matrix E3 = (2 1)
(0 1)
4. Create an elementary matrix E4 = (1 0)
(3 1)
5. Multiply E3 and E4 to get A = (2 1) (6 4)
Therefore, A can be expressed as a product of elementary matrices, E1 x E2 x E4 = (2 0) (0 1) (1 0) (3 1).
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In Exercises 1 through 4, show that AX = B is equivalent to the upper-triangular system UX = Y and find the solution.
2x1 - 2x2 + 5x3 = 6
2x1 + 3x2 + x3 = 13
- X1 + 4x2 - 4x3 = 3
2x - 2x2 + 5x3 = 6 5x2 - 4x3 = 7 0.9x3 = 1.8
To show that the system AX = B is equivalent to the upper-triangular system UX = Y and find the solution, we need to use the Gaussian elimination method. The Gaussian elimination method is a process of reducing a system of linear equations to an equivalent upper-triangular system.
Step 1: Write the given system of equations in matrix form:
```
| 2 -2 5 | | x1 | | 6 |
| 2 3 1 | * | x2 | = | 13 |
|-1 4 -4 | | x3 | | 3 |
```
Step 2: Use the Gaussian elimination method to reduce the matrix to an upper-triangular form:
```
| 2 -2 5 | | 6 |
| 0 7 -9 | = | 7 |
| 0 0 9/7| | 18/7|
```
Step 3: Write the upper-triangular system in equation form:
```
2x1 - 2x2 + 5x3 = 6
0x1 + 7x2 - 9x3 = 7
0x1 + 0x2 + 9/7x3 = 18/7
```
Step 4: Solve the system using back substitution:
```
x3 = 18/7 * 7/9 = 2
x2 = (7 + 9*2)/7 = 4
x1 = (6 + 2*2 - 5*2)/2 = 1
```
Step 5: Write the solution in vector form:
```
| x1 | | 1 |
| x2 | = | 4 |
| x3 | | 2 |
```
Therefore, the solution to the system AX = B is x1 = 1, x2 = 4, and x3 = 2.
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Graph the solution to the following system of inequalities.
2x+3y<9
Y>_ - 2/3x-4
Then give the coordinates of one point in the solution set.
Point in the solution sets D
The graph of 2x + 3y < 9 and y≥ - 2/3x-4 is given in the attachment.
What is Inequality?A relationship between two expressions or values that are not equal to each other is called 'inequality.
For the first inequality, 2x + 3y < 9, we will start by graphing the line 2x + 3y = 9, which is the boundary line of the inequality.
To do this, we will solve for y:
2x + 3y = 9
3y = -2x + 9
Divide both sides by 3
y = (-2/3)x + 3
For the second inequality, Y≥ - 2/3x-4
Hence, the graph of 2x + 3y < 9 and y≥ - 2/3x-4 is given in the attachment.
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