Answer:
(A): Slope = -1
(B): y = -x + 1
(C): x-intercept as an ordered pair: (1, 0)
(D): y-intercept as an ordered pair: (0, 1)
(E): Equation of a second parallel line: y = -x + 6
(F): Equation of a third perpendicular line: y = x + 2
Explanation:
(A):
Given two points on a line, we can find the slope using the slope formula, which is given by:
m = (y2 - y1) / (x2 - x1), where
- m is the slope,
- (x1, y1) is one point on the line,
- and (x2, y2) is another point on the line.
Thus, we can plug in (4, -3) for (x1, y1) and (6, -5) for (x2, y2) to find m, the slope of the line containing the points:
m = (-5 - (-3)) / (6 - 4)
m = (-5 + 3) / (2)
m = (-2) / (2)
m = -1
Thus, the slope of the line is -1.
(B):
The general equation of the slope-intercept form is given by:
y = mx + b, where
- (x, y) are any point on the line,
- m is the slope,
- b is the y-intercept.
Finding the y-intercept:
We can find the y-intercept by plugging in (4, -3) for (x, y) and -1 for m:
-3 = -1(4) + b
-3 = -4 + b
1 = b
Thus, the equation of the line in slope-intercept form is y = -x + 1.
(C):
Finding the x-coordinate of the x-intercept:
- For any x-intercept, the y-coordinate will always be 0 since the line is intersecting the x-axis.
Thus, we can find the x-coordinate of the x-intercept by plugging in 0 for y and solving for x:
0 = -x + 1
-1 = -x
1 = x
Thus, the x-intercept as an ordered pair is (1, 0).
(D):
Finding the y-coordinate of the y-intercept:
- Similarly, the x-coordinate of any y-intercept will always be 0 since the line is intersecting the y-axis.
Thus, we can find the y-coordinate of the y-intercept by plugging in 0 for x and solving for y:
y = -0 + 1
y = 1
Thus, the y-intercept as an ordered pair is (0, 1).
(E):
Find the slope of the line:
- The equations of parallel lines always have the same slope.
Thus, the slope of the other line is also -1.
Finding the y-intercept of the line:
We can find b, the y-intercept, of the other line by plugging in (6, 0) for (x, y) and -1 for m:
0 = -1(6) + b
0 = -6 + b
6 = b
Thus, the equation of a second line that is parallel to the line that passes through (6, 0) is y = -x + 6.
(F):
Finding the slope of the line:
The slopes of perpendicular lines are negative reciprocals of each other as shown by the following formula:
m2 = -1 / m1, where
- m2 is the slope of the line we're trying to find,
- and m1 is the slope of the line we're given.
Thus, we plug in -1 for m1 to find m2, the slope of the other line:
m2 = -1 / -1
m = 1
Finding the y-intercept of the other line:
Now we can plug in (2, 4) for (x, y) and 1 for m to find b, the y-intercept of the line:
4 = 1(2) + b
4 = 2 + b
2 = b
Thus, the equation of a third line that is perpendicular to the line that passes through (2, 4) is y = x + 2.