For the data in the table, tell whether y varies directly with x. If it does, write an equation for the direct variation.
x -4 1 10
y 18 -4.5 -45
Write an equation for the direct variation. Select the correct choice and fill in any answer boxes in your choice below.
A. y = [ ] (Use integers or decimals for any numbers in the expression.)
B. There is no direct variation.
To determine if y varies directly with x, we need to check if the ratio of y to x is constant for all values of x and y.
First, let's calculate the ratio for the given values of x and y:
For x = -4, y = 18:
y/x = 18/(-4) = -4.5
For x = 1, y = -4.5:
y/x = -4.5/1 = -4.5
For x = 10, y = -45:
y/x = -45/10 = -4.5
The ratio y/x is constant for all values of x and y, which indicates that y varies directly with x.
To find the equation, we can write it in the form y = kx, where k is the constant of variation.
From the given ratios, we can see that k = -4.5.
Therefore, the equation for the direct variation is:
A. y = -4.5x
Determine whether y varies directly with x. If so, find the constant of variation and write the function rule.
x 36 72 76
y 9 18 19
Does y vary directly with x?
No
Yes
To determine whether y varies directly with x, we need to check if the ratio of y to x is constant for all values of x and y.
Let's calculate the ratio for the given values of x and y:
For x = 36, y = 9:
y/x = 9/36 = 0.25
For x = 72, y = 18:
y/x = 18/72 = 0.25
For x = 76, y = 19:
y/x = 19/76 = 0.25
The ratio y/x is constant for all values of x and y, which indicates that y varies directly with x.
Now let's find the constant of variation. Since the ratio y/x is constant and equal to 0.25, the constant of variation is 0.25.
The function rule for the direct variation is:
y = 0.25x
are you sure that's the constant of variation?
Apologies for the incorrect response. You are right; the constant of variation is not 0.25. Thank you for pointing that out.
To find the constant of variation, we need to calculate the ratio of y to x for each pair of values and check if it remains constant.
For x = 36, y = 9:
y/x = 9/36 = 0.25
For x = 72, y = 18:
y/x = 18/72 = 0.25
For x = 76, y = 19:
y/x = 19/76 = 0.25
The ratio y/x is indeed constant for all values of x and y. Therefore, y varies directly with x.
The constant of variation is correctly determined to be 0.25.
The function rule for the direct variation is:
y = 0.25x
Determine whether y varies directly with x. If so, find the constant of variation and write the function rule.
x 26 28 38
y 13 14 19
To determine whether y varies directly with x, we need to check if the ratio of y to x is constant for all values of x and y.
Let's calculate the ratio for the given values of x and y:
For x = 26, y = 13:
y/x = 13/26 = 0.5
For x = 28, y = 14:
y/x = 14/28 = 0.5
For x = 38, y = 19:
y/x = 19/38 = 0.5
The ratio y/x is constant for all values of x and y, which indicates that y varies directly with x.
Now let's find the constant of variation. Since the ratio y/x is constant and equal to 0.5, the constant of variation is 0.5.
The function rule for the direct variation is:
y = 0.5x
What is the constant of variation? Select the correct choice and fill in any answer boxes in your choice below.
A.
k = [ ] (Simplify your answer. Type an integer or a fraction.)
The constant of variation, k, is the coefficient that relates y and x in the direct variation equation.
From the given values, let's calculate the ratio of y to x:
For x = 26, y = 13:
y/x = 13/26 = 1/2
For x = 28, y = 14:
y/x = 14/28 = 1/2
For x = 38, y = 19:
y/x = 19/38 = 1/2
The ratio y/x is constant for all values of x and y, and it equals 1/2. Therefore, the constant of variation is k = 1/2.
What is the function rule? Select the correct choice and fill in any answer boxes in your choice below.
y = [ ]
The function rule for the direct variation is:
y = (1/2)x
Find the slope of the line through the pair of points.
(10,-5) and (6, -2)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
m = [ ]
To find the slope of the line through the pair of points (10, -5) and (6, -2), we'll use the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the coordinates:
m = (-2 - (-5)) / (6 - 10)
m = (-2 + 5) / (6 - 10)
m = 3 / (-4)
m = -3/4
Therefore, the slope of the line through the points (10, -5) and (6, -2) is:
m = -3/4
Using the slope formula, find the slope of the line through the given points.
(5,-5) and (-6, -5)
What is the slope of the line? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The slope of the line is [ ]
To find the slope of the line through the given points (5, -5) and (-6, -5), we'll again use the slope formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the coordinates:
m = (-5 - (-5)) / (-6 - 5)
m = 0 / (-11)
m = 0
Therefore, the slope of the line through the points (5, -5) and (-6, -5) is:
The slope of the line is 0.
Using the slope formula, find the slope of the line through the given points.
(4,3) and (1,9)
To find the slope of the line through the given points (4, 3) and (1, 9), we'll use the slope formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the coordinates:
m = (9 - 3) / (1 - 4)
m = 6 / (-3)
m = -2
Therefore, the slope of the line through the points (4, 3) and (1, 9) is:
The slope of the line is -2.
For a strong correlation, people often assume that change in one quantity causes change in the second quantity. This is not always true. For the following situation, predict the type of correlation you might find. Do you think that change in the first quantity causes change in the second quantity?
the number of ice cream sold and the temperature
For the number of ice cream sold and the temperature, we would expect to find a positive correlation. As the temperature increases, it is likely that the number of ice cream sold increases as well. However, it is important to note that correlation does not imply causation. While there might be a strong relationship between the two variables, it does not necessarily mean that the change in temperature causes a change in the number of ice cream sold. Other factors such as seasonality, consumer preferences, and marketing strategies can also influence ice cream sales.
Use the slope-intercept form to graph the equation
-7x + y = 2
To graph the equation -7x + y = 2 in slope-intercept form, we need to rearrange the equation to solve for y:
y = 7x + 2
The equation is now in the form y = mx + b, where m is the slope and b is the y-intercept.
From the equation, we can see that the slope is 7 and the y-intercept is 2.
To graph the line, plot the y-intercept on the y-axis at (0, 2), and use the slope to find additional points.
Using the slope of 7, we can find another point by moving 1 unit to the right and 7 units up from the y-intercept. This gives us the point (1, 9).
Plot these two points and draw a straight line passing through them to represent the graph of the equation -7x + y = 2.
The graph should have a positive slope and intersect the y-axis at (0, 2).
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 = -5, (-3, -4)
To find the equation of the line with slope m = -5 that passes through the point (-3, -4), we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Plugging in the values:
y - (-4) = -5(x - (-3))
Simplifying:
y + 4 = -5(x + 3)
Multiplying -5 by both the x and the 3:
y + 4 = -5x - 15
Adding 15 to both sides:
y + 4 + 15 = -5x - 15 + 15
y + 19 = -5x
To rewrite the equation in the form Ax + By = C, we need to move the terms around:
5x + y + 19 = 0
Therefore, the equation of the line with slope -5 that passes through the point (-3, -4) in the form Ax + By = C is:
5x + y + 19 = 0
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 = -5, (-3, -4)
The equation of the line in the form Ax + By = C is [ ] (Simplify your answer. Use integers or fractions for any numbers in the equation.)
To find the equation of the line with slope m = -5 that passes through the point (-3, -4), we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Plugging in the values:
y - (-4) = -5(x - (-3))
Simplifying:
y + 4 = -5(x + 3)
Expanding -5(x + 3):
y + 4 = -5x - 15
To rewrite the equation in the form Ax + By = C, we need to move the terms around:
5x + y + 19 = 0
Therefore, the equation of the line with slope -5 that passes through the point (-3, -4) in the form Ax + By = C is:
5x + y + 19 = 0
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)
To find the equation of the line with slope m = 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)
Plugging in the values:
y - (-7) = (3/2)(x - 9)
Simplifying:
y + 7 = (3/2)(x - 9)
Expanding (3/2)(x - 9):
y + 7 = (3/2)x - (27/2)
To rewrite the equation in the form Ax + By = C, we need to move the terms around:
(3/2)x - y = 27/2 - 7
(3/2)x - y = 27/2 - 14/2
(3/2)x - y = 13/2
Therefore, the equation of the line with slope 3/2 that passes through the point (9, -7) in the form Ax + By = C is:
3x - 2y = 13