This semester you have learnt how to solve simultaneous equations algebraically. Another method of solving these equations is using Cramer's Rule. Only consider situations where there are the same number of variables and equations. (ii) Give an example of solving a two variable and a three variable set of simultaneous equations using Cramer's Rule with an explanation of the steps involved.

Question
Matrices
asked 2020-11-16
This semester you have learnt how to solve simultaneous equations algebraically. Another method of solving these equations is using Cramer's Rule. Only consider situations where there are the same number of variables and equations.
(ii) Give an example of solving a two variable and a three variable set of simultaneous equations using Cramer's Rule with an explanation of the steps involved.

Answers (1)

2020-11-17
Step 1
Here we use Cramer's rule to find the solution of system of linear equations. We find determinants of matrices.
Step 2
Answer: two variables
12x+3y=15
2x-3y=13
Find the determinant of matrices
\(\Rightarrow D=\begin{vmatrix}12 & 3 \\2 & -3 \end{vmatrix}=-36-6=-42\)
\(D_x=\begin{vmatrix}15 & 3 \\13 & -3 \end{vmatrix}=-45-39=-84\)
\(D_y=\begin{vmatrix}12 & 15 \\2 & 13 \end{vmatrix}=156-30=126\)
Now the solution is: \(x=\frac{D_x}{D}=\frac{-84}{-42}=2\)
\(y=\frac{D_y}{D}=\frac{126}{-42}=-3\)
\(\Rightarrow (x,y)=(2,-3)\)
(2) Three variables:
\(x+y-z=6\)
\(3x-2y+z=-5\)
\(x+3y-2z=14\)
\(D=\begin{vmatrix}1 & 1&-1 \\3 & -2&1\\1&3&-2 \end{vmatrix}=-3\)
\(D_x=\begin{vmatrix}6 & 1&-1 \\-5 & -2&1\\14&3&-2 \end{vmatrix}=-3\)
\(D_y=\begin{vmatrix}1 & 6&-1 \\3 & -5&1\\1&14&-2 \end{vmatrix}=-9\)
\(D_z=\begin{vmatrix}1 & 1&6 \\3 & -2&-5\\1&3&14 \end{vmatrix}=6\)
Solutions are: \(x=\frac{D_x}{D}=\frac{-3}{-3}=1\)
\(y=\frac{D_y}{D}=\frac{-9}{-3} , z=\frac{D_z}{D}=\frac{6}{-3}=-2\)
\(\Rightarrow (x,y,z)=(1,3,-2)\)
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Relevant Questions

asked 2020-12-25
Case: Dr. Jung’s Diamonds Selection
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After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
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8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
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asked 2020-10-23
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
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asked 2021-02-25
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In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
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4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
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How could you use the pattern in that “third row” to figure out the actual rule for the exponential growth or decay model?
asked 2021-01-10
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asked 2021-02-16
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1.What do the unknowns in this system represent and what are the two equations that that need to be solved?
2.Finally, solve the system of equations.
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