Question
Find an equation of the line with the given slope that passes through the given point. Write the equation in the form Ax + By = C
m = 3/2, (9, -7)
m = 3/2, (9, -7)
Answers
Answer
Write the equation of the line in slope-intercept form given the slope and the coordinates of the y-intercept.
m = 6/5; {0,4/5}
m = 6/5; {0,4/5}
Answer
Write the equation of the line in slope-intercept form given the slope and the coordinates of the y-intercept.
m = 6/5; {0,4/9}
m = 6/5; {0,4/9}
Answer
Find an equation of the line described below. Write the equation in slope-intercept form (solved for y), when possible.
Through (10,4) and (4,10)
What is the equation of the line?
[ ] (Simplify your answer.)
Through (10,4) and (4,10)
What is the equation of the line?
[ ] (Simplify your answer.)
Answer
Find an equation of the line described below. Write the equation in slope-intercept form (solved for y), when possible.
With undefined slope, through (-1/2, -4)
What is the equation of the line?
[ ] (Simplify your answer.)
With undefined slope, through (-1/2, -4)
What is the equation of the line?
[ ] (Simplify your answer.)
Answer
Write an equation in standard form of the line that contains the point (-5,2) and is
a. parallel to the line y = 5x + 13
b. perpendicular to the line y = 5x + 13
Which of the following equations, written in standard form, is parallel to the line y = 5x + 13 and contains the point (-5,2) ? Choose the correct answer below.
A. 5x + y = 27
B. x - y = -7/5
C. 5x - y = - 27
D. x + y = 7/5
a. parallel to the line y = 5x + 13
b. perpendicular to the line y = 5x + 13
Which of the following equations, written in standard form, is parallel to the line y = 5x + 13 and contains the point (-5,2) ? Choose the correct answer below.
A. 5x + y = 27
B. x - y = -7/5
C. 5x - y = - 27
D. x + y = 7/5
Answer
Write an equation in standard form of the line that contains the point (1,-6) and is
a. parallel to the line 4x + 9y = 2
b. perpendicular to the line 4x + 9y = 2
Which of the following equations, written in standard form, is parallel to the line 4x + 9y = 2 and contains the point (1,-6) ? Choose the correct answer below.
a. 9x - 4y = 15
b. 9x - 4y = 50
c. 4x + 9y = -50
d. 4x + 9y = -15
a. parallel to the line 4x + 9y = 2
b. perpendicular to the line 4x + 9y = 2
Which of the following equations, written in standard form, is parallel to the line 4x + 9y = 2 and contains the point (1,-6) ? Choose the correct answer below.
a. 9x - 4y = 15
b. 9x - 4y = 50
c. 4x + 9y = -50
d. 4x + 9y = -15
Answer
b. Which of the following equations, written in standard form, is perpendicular to the line 4x + 9y = 2 and contains the point (1,-6) ? Choose the correct answer below.
9x - 4y = 58
9x - 4y= 33
4x + 9y = -33
4x + 9y = -58
9x - 4y = 58
9x - 4y= 33
4x + 9y = -33
4x + 9y = -58
Answer
9x - 4y = 58
9x - 4y= 33
4x + 9y = -33
4x + 9y = -58
(-9x + 4y = -33 wasn't a answer on this list)
9x - 4y= 33
4x + 9y = -33
4x + 9y = -58
(-9x + 4y = -33 wasn't a answer on this list)
Answer
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 183.9 | 186.2|187.8|189.0 |190.0|190.8|
(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.
(a) The linear model for the data is y = [ ] x + [ ]. (Round to two decimal places as needed.)
The table below gives the total sales (in billions of dollars) for the aerospace industry.
Year 2006 | 2007 | 2008 | 2009 | 2010| 2011|
Total Sales 183.9 | 186.2|187.8|189.0 |190.0|190.8|
(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.
(a) The linear model for the data is y = [ ] x + [ ]. (Round to two decimal places as needed.)
Answer
that was incorrect
Answered by
GPT 3.5
To find the equation of the line with slope 3/2 that passes through the point (9, -7), we can use the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope.
Plugging in the values, we have:
y - (-7) = (3/2)(x - 9)
Simplifying, we get:
y + 7 = (3/2)(x - 9)
Multiplying through by 2 to get rid of the fraction:
2(y + 7) = 3(x - 9)
Expanding:
2y + 14 = 3x - 27
Rearranging to the standard form Ax + By = C:
3x - 2y = 41
So the equation of the line with slope 3/2 that passes through the point (9, -7) is 3x - 2y = 41.
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope.
Plugging in the values, we have:
y - (-7) = (3/2)(x - 9)
Simplifying, we get:
y + 7 = (3/2)(x - 9)
Multiplying through by 2 to get rid of the fraction:
2(y + 7) = 3(x - 9)
Expanding:
2y + 14 = 3x - 27
Rearranging to the standard form Ax + By = C:
3x - 2y = 41
So the equation of the line with slope 3/2 that passes through the point (9, -7) is 3x - 2y = 41.
Answered by
GPT 3.5
To find the equation of the line in slope-intercept form, we can use the formula:
y = mx + b
where m is the slope and b is the y-intercept.
Given that the slope (m) is 6/5 and the y-intercept is (0, 4/5), we can substitute these values into the equation:
y = (6/5)x + 4/5
This is the equation of the line in slope-intercept form.
y = mx + b
where m is the slope and b is the y-intercept.
Given that the slope (m) is 6/5 and the y-intercept is (0, 4/5), we can substitute these values into the equation:
y = (6/5)x + 4/5
This is the equation of the line in slope-intercept form.
Answered by
GPT 3.5
To find the equation of the line in slope-intercept form, we can use the formula:
y = mx + b
where m is the slope and b is the y-intercept.
Given that the slope (m) is 6/5 and the y-intercept is (0, 4/9), we can substitute these values into the equation:
y = (6/5)x + 4/9
This is the equation of the line in slope-intercept form.
y = mx + b
where m is the slope and b is the y-intercept.
Given that the slope (m) is 6/5 and the y-intercept is (0, 4/9), we can substitute these values into the equation:
y = (6/5)x + 4/9
This is the equation of the line in slope-intercept form.
Answered by
GPT 3.5
To find the equation of the line passing through the points (10,4) and (4,10), we can use the point-slope form of a linear equation:
(y - y₁) = m(x - x₁)
where (x₁, y₁) are the coordinates of one of the given points, and m is the slope.
First, we need to find the slope (m) using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the coordinates of the points (10,4) and (4,10):
m = (10 - 4) / (4 - 10)
m = 6 / -6
m = -1
Now we can choose either point to substitute into the point-slope form. Let's use the point (10,4):
(y - 4) = -1(x - 10)
Simplifying:
y - 4 = -x + 10
To write the equation in slope-intercept form, we can solve for y:
y = -x + 14
So the equation of the line passing through the points (10,4) and (4,10) is y = -x + 14.
(y - y₁) = m(x - x₁)
where (x₁, y₁) are the coordinates of one of the given points, and m is the slope.
First, we need to find the slope (m) using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the coordinates of the points (10,4) and (4,10):
m = (10 - 4) / (4 - 10)
m = 6 / -6
m = -1
Now we can choose either point to substitute into the point-slope form. Let's use the point (10,4):
(y - 4) = -1(x - 10)
Simplifying:
y - 4 = -x + 10
To write the equation in slope-intercept form, we can solve for y:
y = -x + 14
So the equation of the line passing through the points (10,4) and (4,10) is y = -x + 14.
Answered by
GPT 3.5
If the slope is undefined, it means that the line is vertical. A vertical line has the equation of the form x = a, where "a" is the x-coordinate of any point on the line. In this case, the line passes through the point (-1/2, -4), so the equation of the line is x = -1/2.
Since we are asked to write the equation in slope-intercept form (solved for y), we can't directly convert the equation x = -1/2 to slope-intercept form, as there is no y-term.
Since we are asked to write the equation in slope-intercept form (solved for y), we can't directly convert the equation x = -1/2 to slope-intercept form, as there is no y-term.
Answered by
GPT 3.5
a. To find a line parallel to the equation y = 5x + 13, we know that parallel lines have the same slope. So the slope of our desired line will also be 5. Using the point-slope form, we can find the equation:
(y - y₁) = m(x - x₁)
Plugging in the values (x₁, y₁) = (-5, 2) and m = 5, we have:
(y - 2) = 5(x - (-5))
(y - 2) = 5(x + 5)
y - 2 = 5x + 25
5x - y = -23
Therefore, the line parallel to y = 5x + 13 and containing the point (-5, 2) is represented by the equation 5x - y = -23.
b. To find a line perpendicular to the equation y = 5x + 13, we know that perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is 5, so the perpendicular line will have a slope of -1/5. Using the point-slope form and plugging in the values (x₁, y₁) = (-5, 2) and m = -1/5:
(y - 2) = (-1/5)(x - (-5))
(y - 2) = (-1/5)(x + 5)
y - 2 = (-1/5)x - 1
x + 5y = 7
Therefore, the line perpendicular to y = 5x + 13 and containing the point (-5, 2) is represented by the equation x + 5y = 7.
The correct answer for the equation in standard form that is parallel to the line y = 5x + 13 and contains the point (-5,2) is option B: 5x - y = -27.
(y - y₁) = m(x - x₁)
Plugging in the values (x₁, y₁) = (-5, 2) and m = 5, we have:
(y - 2) = 5(x - (-5))
(y - 2) = 5(x + 5)
y - 2 = 5x + 25
5x - y = -23
Therefore, the line parallel to y = 5x + 13 and containing the point (-5, 2) is represented by the equation 5x - y = -23.
b. To find a line perpendicular to the equation y = 5x + 13, we know that perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is 5, so the perpendicular line will have a slope of -1/5. Using the point-slope form and plugging in the values (x₁, y₁) = (-5, 2) and m = -1/5:
(y - 2) = (-1/5)(x - (-5))
(y - 2) = (-1/5)(x + 5)
y - 2 = (-1/5)x - 1
x + 5y = 7
Therefore, the line perpendicular to y = 5x + 13 and containing the point (-5, 2) is represented by the equation x + 5y = 7.
The correct answer for the equation in standard form that is parallel to the line y = 5x + 13 and contains the point (-5,2) is option B: 5x - y = -27.
Answered by
GPT 3.5
a. To find a line parallel to the equation 4x + 9y = 2, we need to maintain the same slope. To do this, we can rewrite the equation in slope-intercept form (y = mx + b):
9y = -4x + 2
y = (-4/9)x + 2/9
Since the parallel line has the same slope, we know that the equation will also be in the form y = (-4/9)x + b. To find the value of b, we can substitute the coordinates of the given point (1, -6):
-6 = (-4/9)(1) + b
-6 = -4/9 + b
b = -6 + 4/9
b = -54/9 + 4/9
b = -50/9
So, the equation of the line that is parallel to 4x + 9y = 2 and contains the point (1, -6) is y = (-4/9)x - (50/9).
If we rewrite this equation in standard form, we multiply by 9 to clear the fraction:
9y = -4x - 50
4x + 9y = -50
Therefore, the correct answer is option c: 4x + 9y = -50.
9y = -4x + 2
y = (-4/9)x + 2/9
Since the parallel line has the same slope, we know that the equation will also be in the form y = (-4/9)x + b. To find the value of b, we can substitute the coordinates of the given point (1, -6):
-6 = (-4/9)(1) + b
-6 = -4/9 + b
b = -6 + 4/9
b = -54/9 + 4/9
b = -50/9
So, the equation of the line that is parallel to 4x + 9y = 2 and contains the point (1, -6) is y = (-4/9)x - (50/9).
If we rewrite this equation in standard form, we multiply by 9 to clear the fraction:
9y = -4x - 50
4x + 9y = -50
Therefore, the correct answer is option c: 4x + 9y = -50.
Answered by
GPT 3.5
To find a line perpendicular to the equation 4x + 9y = 2, we need to find the negative reciprocal of the slope. The given equation is already in standard form, so we can rewrite it in slope-intercept form:
9y = -4x + 2
y = (-4/9)x + 2/9
The slope of the given line is -4/9. The negative reciprocal of that slope is 9/4.
Now, we can write the equation of the line in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. We can use the point (1, -6) to find the value of b:
-6 = (9/4)(1) + b
-6 = 9/4 + b
b = -6 - 9/4
b = -33/4
Therefore, the equation of the line that is perpendicular to 4x + 9y = 2 and contains the point (1, -6) is y = (9/4)x - (33/4).
If we rewrite this equation in standard form, we can clear the fractions by multiplying through by 4:
4y = 9x - 33
-9x + 4y = -33
Therefore, the correct answer is option c: -9x + 4y = -33.
9y = -4x + 2
y = (-4/9)x + 2/9
The slope of the given line is -4/9. The negative reciprocal of that slope is 9/4.
Now, we can write the equation of the line in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. We can use the point (1, -6) to find the value of b:
-6 = (9/4)(1) + b
-6 = 9/4 + b
b = -6 - 9/4
b = -33/4
Therefore, the equation of the line that is perpendicular to 4x + 9y = 2 and contains the point (1, -6) is y = (9/4)x - (33/4).
If we rewrite this equation in standard form, we can clear the fractions by multiplying through by 4:
4y = 9x - 33
-9x + 4y = -33
Therefore, the correct answer is option c: -9x + 4y = -33.
Answered by
GPT 3.5
Apologies for the error. Given the options provided, the equation in standard form that is perpendicular to the line 4x + 9y = 2 and contains the point (1,-6) is:
4x + 9y = -58
So the correct answer is option (d).
4x + 9y = -58
So the correct answer is option (d).
Answered by
GPT 3.5
To find the linear model for the data, we will use the least-squares regression method to find the equation of the line that best fits the given data points.
Step 1: Calculate the mean for both the x and y values.
x̄ = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 2008.5
ȳ = (183.9 + 186.2 + 187.8 + 189.0 + 190.0 + 190.8) / 6 = 187.95
Step 2: Calculate the deviations for each x and y value from their respective means and their product.
(x - x̄) and (y - ȳ), and (x - x̄)(y - ȳ):
For x values:
2006 - 2008.5 = -2.5
2007 - 2008.5 = -1.5
2008 - 2008.5 = -0.5
2009 - 2008.5 = 0.5
2010 - 2008.5 = 1.5
2011 - 2008.5 = 2.5
For y values:
183.9 - 187.95 = -4.05
186.2 - 187.95 = -1.75
187.8 - 187.95 = -0.15
189.0 - 187.95 = 1.05
190.0 - 187.95 = 2.05
190.8 - 187.95 = 2.85
For the product of the deviations:
(-2.5) * (-4.05) = 10.125
(-1.5) * (-1.75) = 2.625
(-0.5) * (-0.15) = 0.075
(0.5) * (1.05) = 0.525
(1.5) * (2.05) = 3.075
(2.5) * (2.85) = 7.125
Step 3: Calculate the sum of the squared deviations for x and y, as well as the sum of the product of deviations.
Σ(x - x̄)² = (-2.5)² + (-1.5)² + (-0.5)² + (0.5)² + (1.5)² + (2.5)² = 14.5
Σ(y - ȳ)² = (-4.05)² + (-1.75)² + (-0.15)² + (1.05)² + (2.05)² + (2.85)² = 31.3375
Σ(x - x̄)(y - ȳ) = 10.125 + 2.625 + 0.075 + 0.525 + 3.075 + 7.125 = 23.55
Step 4: Calculate the slope:
b = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)²
b = 23.55 / 14.5 = 1.62069
Step 5: Calculate the y-intercept:
a = ȳ - b * x̄
a = 187.95 - 1.62069 * 2008.5 = -2507.681745
The linear model for the data is:
y = 1.62x - 2507.68
So, the linear model for the data is y = 1.62x - 2507.68. (Round to two decimal places if necessary.)
Step 1: Calculate the mean for both the x and y values.
x̄ = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 2008.5
ȳ = (183.9 + 186.2 + 187.8 + 189.0 + 190.0 + 190.8) / 6 = 187.95
Step 2: Calculate the deviations for each x and y value from their respective means and their product.
(x - x̄) and (y - ȳ), and (x - x̄)(y - ȳ):
For x values:
2006 - 2008.5 = -2.5
2007 - 2008.5 = -1.5
2008 - 2008.5 = -0.5
2009 - 2008.5 = 0.5
2010 - 2008.5 = 1.5
2011 - 2008.5 = 2.5
For y values:
183.9 - 187.95 = -4.05
186.2 - 187.95 = -1.75
187.8 - 187.95 = -0.15
189.0 - 187.95 = 1.05
190.0 - 187.95 = 2.05
190.8 - 187.95 = 2.85
For the product of the deviations:
(-2.5) * (-4.05) = 10.125
(-1.5) * (-1.75) = 2.625
(-0.5) * (-0.15) = 0.075
(0.5) * (1.05) = 0.525
(1.5) * (2.05) = 3.075
(2.5) * (2.85) = 7.125
Step 3: Calculate the sum of the squared deviations for x and y, as well as the sum of the product of deviations.
Σ(x - x̄)² = (-2.5)² + (-1.5)² + (-0.5)² + (0.5)² + (1.5)² + (2.5)² = 14.5
Σ(y - ȳ)² = (-4.05)² + (-1.75)² + (-0.15)² + (1.05)² + (2.05)² + (2.85)² = 31.3375
Σ(x - x̄)(y - ȳ) = 10.125 + 2.625 + 0.075 + 0.525 + 3.075 + 7.125 = 23.55
Step 4: Calculate the slope:
b = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)²
b = 23.55 / 14.5 = 1.62069
Step 5: Calculate the y-intercept:
a = ȳ - b * x̄
a = 187.95 - 1.62069 * 2008.5 = -2507.681745
The linear model for the data is:
y = 1.62x - 2507.68
So, the linear model for the data is y = 1.62x - 2507.68. (Round to two decimal places if necessary.)
Answered by
GPT 3.5
Apologies for the incorrect response. Let's recalculate the linear model using the least-squares regression method.
Step 1: Calculate the mean for both the x and y values.
x̄ = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 2008.5
ȳ = (183.9 + 186.2 + 187.8 + 189.0 + 190.0 + 190.8) / 6 = 188.8
Step 2: Calculate the deviations for each x and y value from their respective means and their product.
(x - x̄) and (y - ȳ), and (x - x̄)(y - ȳ):
For x values:
2006 - 2008.5 = -2.5
2007 - 2008.5 = -1.5
2008 - 2008.5 = -0.5
2009 - 2008.5 = 0.5
2010 - 2008.5 = 1.5
2011 - 2008.5 = 2.5
For y values:
183.9 - 188.8 = -4.9
186.2 - 188.8 = -2.6
187.8 - 188.8 = -1.0
189.0 - 188.8 = 0.2
190.0 - 188.8 = 1.2
190.8 - 188.8 = 2.0
For the product of the deviations:
(-2.5) * (-4.9) = 12.25
(-1.5) * (-2.6) = 3.9
(-0.5) * (-1.0) = 0.5
(0.5) * (0.2) = 0.1
(1.5) * (1.2) = 1.8
(2.5) * (2.0) = 5.0
Step 3: Calculate the sum of the squared deviations for x and y, as well as the sum of the product of deviations.
Σ(x - x̄)² = (-2.5)² + (-1.5)² + (-0.
Step 1: Calculate the mean for both the x and y values.
x̄ = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 2008.5
ȳ = (183.9 + 186.2 + 187.8 + 189.0 + 190.0 + 190.8) / 6 = 188.8
Step 2: Calculate the deviations for each x and y value from their respective means and their product.
(x - x̄) and (y - ȳ), and (x - x̄)(y - ȳ):
For x values:
2006 - 2008.5 = -2.5
2007 - 2008.5 = -1.5
2008 - 2008.5 = -0.5
2009 - 2008.5 = 0.5
2010 - 2008.5 = 1.5
2011 - 2008.5 = 2.5
For y values:
183.9 - 188.8 = -4.9
186.2 - 188.8 = -2.6
187.8 - 188.8 = -1.0
189.0 - 188.8 = 0.2
190.0 - 188.8 = 1.2
190.8 - 188.8 = 2.0
For the product of the deviations:
(-2.5) * (-4.9) = 12.25
(-1.5) * (-2.6) = 3.9
(-0.5) * (-1.0) = 0.5
(0.5) * (0.2) = 0.1
(1.5) * (1.2) = 1.8
(2.5) * (2.0) = 5.0
Step 3: Calculate the sum of the squared deviations for x and y, as well as the sum of the product of deviations.
Σ(x - x̄)² = (-2.5)² + (-1.5)² + (-0.
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