Question

A linear model for the data in the table is shown in the scatter plot.

x 1 2 4 5 6 7 8 9
y 25 24 15 12 11 13 6 4
(a) Which two points should you use to find the equation of the model? Explain.
(b) Use the two points you chose in Part (a) to find the slope of the linear model, leaving your slope in fraction form. Show your work.
(c) What is the equation of the linear model in point-slope form?
(d) Rearrange the equation you wrote in Part (c) into slope-intercept form. Show your work.

Answer 1

To find the equation of a linear model, you would first create a scatter plot, determine two points from the line of best fit, calculate the slope, write the equation in point-slope form, and then rearrange it into slope-intercept form. Without specific points or a regression line, the calculations cannot be provided, but the methodology is clear.

Step-by-step explanation:

To find the equation of a linear model based on the given data, we should follow these steps:

1. Enter the data into a calculator and make a scatter plot.
2. Identify two points that lie close to the line of best fit for ease of calculation. In practice, you would use a calculator's regression function to find the equation of the least-squares regression line and use this line to pick two points. However, hypothetically, if we're selecting two points without calculator aid, we would look for two points that seem to represent the trend of the data well.
3. Use the two chosen points to calculate the slope of the line. The slope (m) is calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.
4. Write the equation of the linear model in point-slope form, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is one of the points used to find the slope.
5. Rearrange the equation from point-slope form into slope-intercept form (y = mx + b) by solving for y.

If using a calculator, after plotting the data, you would use the regression function to compute the precise least-squares line, which is the best fit for the data points. The equation of the least-squares regression line generally has the form ý = a + bx, where a is the y-intercept, and b is the slope.

For parts (b), (c), and (d) of the question, the specific points and calculations cannot be provided without the scatter plot or the regression line. However, the provided methodology explains the steps needed to correctly answer the question once you have access to the actual data or the scatter plot created from it.