# What would be the linear equation for this function, & the graph? (X_1,Y_1)=(4,8), (X_2,Y_2)=(6,-9)

Question
Linear equations and graphs
What would be the linear equation for this function, & the graph?
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left({4},{8}\right)},{\left({X}_{{2}},{Y}_{{2}}\right)}={\left({6},-{9}\right)}$$

2020-10-24
Formula used:
Point-slope form of line:
$$\displaystyle{\left({y}_{{2}}-{y}_{{1}}\right)}={m}{\left({x}_{{2}}-{x}_{{1}}\right)}$$
So, by the Point-Slope form of line the equation of line is given below:
$$\displaystyle{\left(-{9}-{8}\right)}={m}{\left({6}-{4}\right)}$$
$$\displaystyle-{17}={m}\times{2}$$
$$\displaystyle{2}{m}=-{17}$$
$$\displaystyle{m}=-\frac{{17}}{{2}}$$
Now $$\displaystyle{m}=-\frac{{17}}{{2}}$$ and take any points to find the linear equation.
$$\displaystyle{\left({x}_{{1}},{y}_{{1}}\right)}={\left({4},{8}\right)}{\quad\text{and}\quad}{m}=-\frac{{17}}{{2}}$$
$$\displaystyle{\left({y}-{y}_{{1}}\right)}={m}{\left({x}-{x}_{{1}}\right)}$$
$$\displaystyle{\left({y}-{8}\right)}=-\frac{{17}}{{2}}{\left({x}-{4}\right)}$$
$$\displaystyle{\left({y}-{8}\right)}=\frac{{-{17}{x}}}{{2}}+{34}$$
$$\displaystyle{y}=\frac{{-{17}{x}}}{{2}}+{34}+{8}$$
$$\displaystyle{y}=\frac{{-{17}{x}}}{{2}}+{42}$$
$$\displaystyle{y}=\frac{{-{17}{x}+{84}}}{{2}}$$
$$\displaystyle{2}{y}=-{17}{x}_{{84}}$$
So, the graph is given below:

### Relevant Questions

What would be the linear equation for this function, & the graph?
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left(-{7},{3}\right)},{\left({X}_{{2}},{Y}_{{2}}\right)}={\left({6},{3}\right)}$$
What would be the linear equation for this function, & the graph??
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left({3},{6}\right)},{\left({X}_{{2}},{Y}_{{2}}\right)}={\left({3},-{5}\right)}$$
What would be the linear equation for this function, & the graph?
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left(-{3},{4}\right)},{m}=-{2}$$

(10%) In $$R^2$$, there are two sets of coordinate systems, represented by two distinct bases: $$(x_1, y_1)$$ and $$(x_2, y_2)$$. If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: $$2(x_1)^2 -4(x_1)(y_1) + 5(y_1)^2 - 36 = 0$$ and $$(x_2)^2 + 6(y_2)^2 - 36 = 0.$$ Find the transformation matrix between these two coordinate systems: $$(x_1, y_1)$$ and $$(x_2, y_2)$$.

The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?

A graph of a linear equation passes through ( -2,0) and (0,-6) is the $$3x-y=6$$, both ordered pairs solutions for the equation

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
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?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
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?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) 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?
Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If $$\displaystyle{x}_{{h}}$$ is a solution of the system Ax=0, then $$\displaystyle{x}_{{1}}+{x}_{{h}}$$ is a solution of the system Ax=b.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is a solution of the system Ax=0