For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.
X Y
52 39
32 24
2o 15
8 6
Find the slope of the given lines:
2. (6, 5) & (9, 10)
3. (-2, -4) & (-2, -6)
Write the equation of the line in point slope form:
4. (5, -9); m = 6
Write the equation of the line in slope-intercept form:
5. (4, 7); m = (- 3/8)
6. (-5, -5) & (-1, 3)
Write the equation of the line in standard form:
7. (-3, 6); m = (1/2)
8. (1, 0) & (-5, 3)
9. Write the equation of the line perpendicular to 4x – 5y = -10 through the point
(2, 3). Write the equation in slope-intercept form. (2 pts.)
What is the slope of the perpendicular line: ______________________
What is the equation: ________________________________
To determine if y varies directly with x in the given table, we need to check if the ratio of y to x remains constant for all corresponding values. Let's compare the ratios:
1. For x = 52 and y = 39, the ratio y/x is 39/52 = 0.75.
2. For x = 32 and y = 24, the ratio y/x is 24/32 = 0.75.
3. For x = 20 and y = 15, the ratio y/x is 15/20 = 0.75.
4. For x = 8 and y = 6, the ratio y/x is 6/8 = 0.75.
Since the ratio of y to x remains constant at 0.75 for all values, we can conclude that y varies directly with x. This means there is a direct variation relationship between y and x.
To write an equation for direct variation, we can use the formula y = kx, where k represents the constant of variation. To find the value of k, we can choose any set of x and y values and solve for k.
Using the first set of values (52, 39):
39 = k * 52
k = 39/52
k = 0.75
Therefore, the equation for direct variation is y = 0.75x.
Moving on to the next set of questions:
2. To find the slope of a line passing through two given points (x1, y1) and (x2, y2), we can use the formula:
slope = (y2 - y1) / (x2 - x1)
Using (6, 5) and (9, 10):
slope = (10 - 5) / (9 - 6)
= 5 / 3
So, the slope of the line is 5/3.
3. For the given points (-2, -4) and (-2, -6), we notice that the x-coordinates are the same. This indicates a vertical line. The slope of a vertical line is undefined, as the denominator becomes zero in the slope formula.
Therefore, the slope of the line (-2, -4) and (-2, -6) is undefined.
4. To write the equation of a line in point-slope form, we can use the formula:
y - y1 = m(x - x1)
Using the point (5, -9) and slope m = 6:
y - (-9) = 6(x - 5)
y + 9 = 6x - 30
y = 6x - 30 - 9
y = 6x - 39
Thus, the equation of the line in point-slope form is y = 6x - 39.
5. To write the equation of a line in slope-intercept form, we can use the formula:
y = mx + b, where m is the slope and b is the y-intercept.
Using the point (4, 7) and slope m = -3/8:
y = (-3/8) * x + b
Plugging in the coordinates of (4, 7), we can solve for b:
7 = (-3/8) * 4 + b
7 = -3/2 + b
b = 7 + 3/2
b = 17/2
Therefore, the equation of the line in slope-intercept form is y = (-3/8)x + 17/2.
6. To find the slope of a line passing through two given points (-5, -5) and (-1, 3), we can use the formula:
slope = (y2 - y1) / (x2 - x1)
Using (-5, -5) and (-1, 3):
slope = (3 - (-5)) / (-1 - (-5))
= 8 / 4
= 2
Thus, the slope of the line is 2.
7. To write the equation of a line in standard form, we can use the formula:
Ax + By = C, where A, B, and C are constants.
Using the point (-3, 6) and slope m = 1/2:
y - y1 = m(x - x1)
y - 6 = (1/2)(x - (-3))
2y - 12 = x + 3
x - 2y = -15
Hence, the equation of the line in standard form is x - 2y = -15.
8. To find the equation of a line passing through two given points (1, 0) and (-5, 3), we can use the formula:
slope = (y2 - y1) / (x2 - x1)
Using (1, 0) and (-5, 3):
slope = (3 - 0) / (-5 - 1)
= 3 / (-6)
= -1/2
The equation of the line in point-slope form using the point (1, 0) and slope -1/2 is:
y - 0 = (-1/2)(x - 1)
y = (-1/2)x + 1/2
Thus, the equation of the line in slope-intercept form is y = (-1/2)x + 1/2.
9. To find the equation of a line perpendicular to the given line 4x - 5y = -10, we need to find the negative reciprocal of the slope of the given line.
The given line can be rearranged to slope-intercept form:
-5y = -4x - 10
y = (4/5)x + 2
The slope of the given line is 4/5. The negative reciprocal of 4/5 is -5/4, so the slope of the perpendicular line will be -5/4.
Using the point (2, 3), we can write the equation of the perpendicular line in slope-intercept form:
y - 3 = (-5/4)(x - 2)
y - 3 = (-5/4)x + 5/2
y = (-5/4)x + 5/2 + 3
y = (-5/4)x + 5/2 + 6/2
y = (-5/4)x + 11/2
Thus, the equation of the perpendicular line in slope-intercept form is y = (-5/4)x + 11/2.
The slope of the perpendicular line is -5/4 and the equation of the line is y = (-5/4)x + 11/2.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
What are your responses?