parallel and perpendicular lines
write the slope-intercept for an equation of the line that passes through the given point and is perpendicular to the graph of each equation (4-2) y=1/2 +1, ( 2,-3) y= -2/3+4 (-8,-7) y=-x-8 (6,-2) y=-3x-6
I will do the last one, you do the others the same way
point is (6,-2)
line is y = -3x - 6
slopes of perpendicular lines are negative reciprocals of each other.
So the new line is
y =(1/3)x + b
sub in the given point
-2 = (1/3)(6) + b
-2 = 2 + b
-4 = b
new equation: y = (1/3)x - 4
To find the equation of a line that is perpendicular to a given line, you need to determine its slope and then take the negative reciprocal of that slope. The slope-intercept form of a line is given by y = mx + b, where "m" represents the slope and "b" represents the y-intercept.
Let's solve each problem one by one:
1. Given the equation y = 1/2x + 1 and the point (4,-2):
- The slope of the given line is 1/2.
- The slope of the line perpendicular to it will be the negative reciprocal of 1/2, which is -2/1 (or -2).
- Now, we can use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the given point.
- Substituting the values, we get y - (-2) = -2(x - 4).
- Simplifying, we obtain y + 2 = -2x + 8.
- By rearranging the equation in slope-intercept form, we get y = -2x + 6. So, the equation of the line is y = -2x + 6.
2. Given the equation y = -2/3x + 4 and the point (2,-3):
- The slope of the given line is -2/3.
- The slope of the line perpendicular to it will be the negative reciprocal of -2/3, which is 3/2.
- Using the point-slope form, we have y - (-3) = 3/2(x - 2).
- Simplifying yields y + 3 = 3/2x - 3.
- Rearranging the equation, we have y = 3/2x - 6. So, the equation of the line is y = 3/2x - 6.
3. Given the equation y = -x - 8 and the point (-8,-7):
- The slope of the given line is -1.
- The slope of the line perpendicular to it will be the negative reciprocal of -1, which is 1.
- Applying the point-slope form equation, we get y - (-7) = 1(x - (-8)).
- Simplifying, we have y + 7 = x + 8.
- Rearranging, we get y = x + 1. So, the equation of the line is y = x + 1.
4. Given the equation y = -3x - 6 and the point (6,-2):
- The slope of the given line is -3.
- The slope of the line perpendicular to it will be the negative reciprocal of -3, which is 1/3.
- Using the point-slope form, we have y - (-2) = 1/3(x - 6).
- Simplifying, we obtain y + 2 = 1/3x - 2.
- Rearranging, we get y = 1/3x - 4. So, the equation of the line is y = 1/3x - 4.