# What would be the linear equation for this function, & the graph? (X_1,Y_1)=(-7,3), (X_2,Y_2)=(6,3)

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

2020-12-15
This is linear equation with connecting two points.
$$\displaystyle{\left({x}_{{1}},{y}_{{1}}\right)}-{\left(-{7},{3}\right)}$$
$$\displaystyle{\left({x}_{{2}},{y}_{{2}}\right)}={\left({6},{3}\right)}$$
Linear eqution
$$\displaystyle{\left({y}-{y}_{{1}}\right)}=\frac{{{y}_{{2}}-{y}_{{1}}}}{{{x}_{{2}}-{x}_{{1}}}}{\left({x}-{x}_{{1}}\right)}$$
y-3=0 or y=3
This is the graph of y=3.

### Relevant Questions

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?
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left({4},{8}\right)},{\left({X}_{{2}},{Y}_{{2}}\right)}={\left({6},-{9}\right)}$$
What would be the linear equation for this function, & the graph?
$$\displaystyle{\left({X}_{{1}},{Y}_{{1}}\right)}={\left(-{3},{4}\right)},{m}=-{2}$$
A graph of a linear equation passes through ( -2,0) & (0,-6) is the 3x-y=6, both ordered pairs solutions for the equation
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?
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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?
Use what you just learned about negative slopes to solve this problem. A storm moves at a rate of 8 miles per hour. It is 200 miles away from Freeport and headed directly for this town. The equation y = 200 - 8x can be used to represent this function. Identify the slope and y-intercept and explain what they represent.
A line L through the origin in $$\displaystyle\mathbb{R}^{{3}}$$ can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of $$\displaystyle\mathbb{R}^{{3}}{b}{y}{s}{h}{o}{w}\in{g}{t}\hat{{\quad\text{if}\quad}}{v}_{{1}}={\left({x}_{{1}},{y}_{{1}},{z}_{{1}}\right)}{\quad\text{and}\quad}{v}_{{2}}={\left({x}_{{2}},{y}_{{2}},{z}_{{2}}\right)}{a}{r}{e}{p}\oint{s}{o}{n}{L}{\quad\text{and}\quad}{k}{i}{s}{a}{n}{y}{r}{e}{a}{l}\nu{m}{b}{e}{r},{t}{h}{e}{n}{k}{v}_{{1}}{\quad\text{and}\quad}{v}_{{1}}+{v}_{{2}}$$ are also points on L.
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio $$x_1$$ (in$) and the amount spent advertising in the newspaper $$x_2$$ (in $) according to $$y=ax_1+bx_2+c$$ The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. $$\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline 2400 & { 800} & { 36,000} \\ \hline 2000 & { 500} & { 30,000} \\ \hline 3000 & { 1000} & { 44,000} \\ \hline\end{array}$$ a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $$y=ax_1+bx_2+c$$ d) Predict the monthly sales if the grocer spends$250 advertising on the radio and \$500 advertising in the newspaper for a given month.