I had an assignment containing 45 problems, and I am unsure of 4, can you check and see if the answers are correct?
1. Write an equation of the line containing the given point and parallel line
(5,-1), 3x-7y=9
My answer was: y=3/7x-22/7
2. Find the equation of the line having the given slope and containing the given point: m=-6,(7,8)
My answer: y=-6x+50
3. Find the slope-intercept of the line that has the given characteristics: slope 8 and y intercept (0,7)My answer was y=8x-7
4. Find an equation of the line containing the given pair of points:(-5,-1)(-9,-7)
I got: y=3/2x-14/5
Let's check your answers step-by-step:
1. Write an equation of the line containing the given point and parallel line: (5,-1), 3x-7y=9
To find an equation of a line parallel to another line, we need to find the equation with the same slope but a different y-intercept. First, let's rearrange the given equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
3x - 7y = 9
-7y = -3x + 9
y = (3/7)x - (9/7)
The slope of this line is 3/7, so the parallel line will also have a slope of 3/7. To find the equation of the line passing through (5, -1), we can substitute these values into the slope-intercept form:
y = (3/7)x + b
Using the coordinates (5, -1) to solve for b:
-1 = (3/7)(5) + b
-1 = 15/7 + b
b = -1 - 15/7
b = -22/7
Thus, the equation of the line parallel to 3x - 7y = 9 and passing through (5, -1) is:
y = (3/7)x - 22/7
Your answer is correct.
2. Find the equation of the line having the given slope and containing the given point: m=-6, (7,8)
To find the equation of a line given the slope (m) and a point (x1, y1), we can use the point-slope form:
y - y1 = m(x - x1)
Substituting the given values:
y - 8 = -6(x - 7)
y - 8 = -6x + 42
y = -6x + 42 + 8
y = -6x + 50
Your answer is correct.
3. Find the slope-intercept form of the line that has the given characteristics: slope 8 and y-intercept (0,7)
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 8 and the y-intercept is (0,7).
Plugging these values into the equation:
y = 8x + 7
Your answer is correct.
4. Find an equation of the line containing the given pair of points: (-5,-1) and (-9,-7)
To find the equation of a line given two points, we can use the point-slope form or the two-point form. Let's use the point-slope form:
y - y1 = ((y2 - y1) / (x2 - x1))(x - x1)
Using the points (-5, -1) and (-9, -7):
y - (-1) = ((-7 - (-1)) / (-9 - (-5)))(x - (-5))
y + 1 = (-6 / -4)(x + 5)
y + 1 = (3/2)(x + 5)
y + 1 = (3/2)x + 15/2
y = (3/2)x + 15/2 - 1
y = (3/2)x + 15/2 - 2/2
y = (3/2)x + 13/2
Your answer is not correct. The correct equation is y = (3/2)x + 13/2.
I hope this helps! Let me know if you have any further questions.
To verify if the given answers are correct, we can solve each problem step by step.
1. Write an equation of the line containing the given point (5,-1) and parallel line 3x-7y=9.
To find the equation of a line parallel to another line, we know that the slopes of the two lines will be the same. So, first, let's find the slope of the given line. The equation of the line can be rewritten in slope-intercept form (y = mx + b) as follows:
3x - 7y = 9
-7y = -3x + 9
y = (3/7)x - 9/7
The slope of the given line is 3/7. Since the parallel line will have the same slope, we can use the point-slope form (y - y1 = m(x - x1)) to find the equation of the parallel line.
Using the point (5,-1):
y - (-1) = (3/7)(x - 5)
y + 1 = (3/7)(x - 5)
y + 1 = (3/7)x - (15/7)
y = (3/7)x - (15/7) - (7/7)
y = (3/7)x - (22/7)
So, the equation of the line containing the given point and parallel to the line 3x - 7y = 9 is y = (3/7)x - (22/7).
Your answer, y = 3/7x - 22/7, is correct.
2. Find the equation of the line having the given slope -6 and containing the given point (7,8).
We can use the point-slope form (y - y1 = m(x - x1)) to find the equation of the line.
Using the point (7,8) and slope -6:
y - 8 = -6(x - 7)
y - 8 = -6x + 42
y = -6x + 42 + 8
y = -6x + 50
So, the equation of the line with slope -6 and containing the point (7,8) is y = -6x + 50.
Your answer, y = -6x + 50, is correct.
3. Find the slope-intercept form of the line with slope 8 and y-intercept (0,7).
In slope-intercept form, the equation of a line is y = mx + b, where m is the slope and b is the y-intercept.
Given slope (m) = 8 and y-intercept (0,7), we can write the equation as:
y = 8x + b
To find the value of b, we substitute the coordinates of the y-intercept into the equation:
7 = 8(0) + b
7 = b
So, the equation of the line with slope 8 and y-intercept (0,7) is y = 8x + 7.
Your answer, y=8x-7, is incorrect. It should be y = 8x + 7.
4. Find an equation of the line containing the given pair of points: (-5,-1) and (-9,-7).
To find the equation of the line passing through two points, we can use the point-slope form or the slope-intercept form.
Using the point-slope form (y - y1 = m(x - x1)):
Using points (-5,-1) and (-9,-7):
y - (-1) = (7 - (-1))/(-9 - (-5)) (x - (-5))
y + 1 = (7 + 1)/(-9 + 5) (x + 5)
y + 1 = 8/(-4) (x + 5)
y + 1 = -2 (x + 5)
y + 1 = -2x - 10
y = -2x - 10 - 1
y = -2x -11
So, the equation of the line passing through the points (-5,-1) and (-9,-7) is y = -2x - 11.
Your answer, y = 3/2x - 14/5, is incorrect. The correct equation is y = -2x -11.
Overall, you got 3 out of 4 answers correct. Great job!