Write the equation of the line in slope-intercept form given the slope and the coordinates of the y-intercept.
m = 6/7;(0, 3/5)
(Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)
The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.
In this case, the slope (m) is 6/7 and the y-intercept is (0, 3/5). Therefore, the equation of the line in slope-intercept form is:
y = (6/7)x + 3/5
Find an equation of the line described below. Write the equation in slope-intercept form (solved for y), when possible.
Through (14,12) and (12,14)
To find the equation of the line passing through the points (14, 12) and (12, 14), we can use the point-slope form of a linear equation.
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (14, 12) and (12, 14), we substitute the values into the formula:
m = (14 - 12) / (12 - 14)
m = 2 / (-2)
m = -1
Now that we have the slope, we can use the point-slope form to write the equation of the line:
y - y1 = m(x - x1)
Using one of the points, let's use (14, 12):
y - 12 = -1(x - 14)
Simplifying, we get:
y - 12 = -x + 14
y = -x + 26
Therefore, the equation of the line in slope-intercept form is:
y = -x + 26
Find an equation of the line described below. Write the equation in slope-intercept form (solved for y), when possible.
With undefined slope, through (3/5,2)
If the slope is undefined, it means that the line is vertical and parallel to the y-axis. In this case, the equation of the line will be of the form x = k, where k is the x-coordinate of any point on the line.
Given that the line passes through (3/5, 2), the equation of the line in slope-intercept form is:
x = 3/5
Write the equation of the line in slope-intercept form given the slope and the coordinates of the y-intercept.
m = 1/4;(0, 2/5)
The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.
In this case, the slope (m) is 1/4 and the y-intercept is (0, 2/5). Therefore, the equation of the line in slope-intercept form is:
y = (1/4)x + 2/5
Find the required linear model using least-squares regression.
The table below gives the total sales (in billions of dollars) for the aerospace industry.
Year: 2006,2007,2008,2009,2010,2011
Total Sales: 178.8,181.1,182.7,183.9,184.9,185.7
(a) Find a linear model for the data with x = 6 corresponding to the year 2006.
(b) Assuming the trend continues, estimate the total sales for the year 2015.
To find the linear model using least-squares regression, we will use the formula:
y = mx + b
where y is the dependent variable (total sales), x is the independent variable (year), m is the slope, and b is the y-intercept.
(a) First, let's calculate the slope (m) and y-intercept (b) using the given data:
Using the formula for the slope:
m = ((n * Σ(xy)) - (Σx * Σy)) / ((n * Σ(x^2)) - (Σx)^2)
where n is the number of data points, Σxy is the sum of the products of x and y, Σx is the sum of x, Σy is the sum of y, and Σ(x^2) is the sum of the squares of x.
In this case, the data is:
Year: 2006, 2007, 2008, 2009, 2010, 2011 (x)
Total Sales: 178.8, 181.1, 182.7, 183.9, 184.9, 185.7 (y)
Calculating the sums:
Σx = 2006 + 2007 + 2008 + 2009 + 2010 + 2011 = 12051
Σy = 178.8 + 181.1 + 182.7 + 183.9 + 184.9 + 185.7 = 1096.1
Σxy = (2006 * 178.8) + (2007 * 181.1) + (2008 * 182.7) + (2009 * 183.9) + (2010 * 184.9) + (2011 * 185.7) = 1209561.4
Σ(x^2) = (2006^2) + (2007^2) + (2008^2) + (2009^2) + (2010^2) + (2011^2) = 24272477
n = 6
Now, we can substitute the values into the formula for slope (m):
m = ((n * Σ(xy)) - (Σx * Σy)) / ((n * Σ(x^2)) - (Σx)^2)
m = ((6 * 1209561.4) - (12051 * 1096.1)) / ((6 * 24272477) - (12051)^2)
m = 6.6450526
Next, let's calculate the y-intercept (b) using the formula:
b = (Σy - m * Σx) / n
b = (1096.1 - 6.6450526 * 12051) / 6
b = -15666.620742
So, the linear model for the data is:
y = 6.6450526x - 15666.620742
(b) To estimate the total sales for the year 2015 (x = 9), we can substitute the value into the linear model equation:
y = 6.6450526 * 2015 - 15666.620742
y = 13370.909715
Therefore, the estimated total sales for the year 2015 is approximately 13370.909715 billion dollars.