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.