Question 632017: Hi, can you help me with this?
i. Find an equation of the line that passes through the point (2, 4) and is perpendicular to the line 3x + 4y -22=0.
ii. Find an equation of the line passing through (-5, -4) and parallel to the line passing through(-3, 2) and (6, 8).
Answer by Maths68(1474)   (Show Source):
You can put this solution on YOUR website! i. Find an equation of the line that passes through the point (2, 4) and is perpendicular to the line 3x + 4y -22=0.
Given
Point (x, y)=(2,4)
Line:
3x+4y-22=0
Rearrange above equation
4y=-3x+22
4y/4=(-3x+22)/4
y=-3x/4+22/4
y=(-3/4)x+11/2
Compare above equation with the equation of line slope-intercept form
y=mx+b
m=-3/4 and b=11/2
Slope of the given line m = -3/4 and y-intercept = b = 11/2
Since required line is perpendicular, the multiplication of the slopes of both lines result in (-1), therefore the slope of the required line will be (4/3)
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Now we have a point (2,4) and slope (4/3) of the required required line we can easily find the required line put these values in the equation of slope-intercept form to find the y-intercept of the required line
y=mx+b
4=(4/3)(2)+b
4=8/3+b
4-8/3=b
(12-8)/3=b
4/3=b
b=4/3
y-intercept of the required line =b=4/3
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Put the values of ‘m’ and ‘b’ in equation of the line slope-intercept form
y=mx+b
y=(4/3)x-4/3
Above equation is the required equation of the line in slope-intercept form.
Graph
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y=(-3/4)x+11/2
y=(4/3)x-4/3
ii. Find an equation of the line passing through (-5, -4) and parallel to the line passing through(-3, 2) and (6, 8).
Points (−3,2) and (6,8)
Use point-slope Equation
m=(y1-y2)/(x1-x2)
m=(2-8)/(-3-6)
m=(-6)/(-9)
m=2/3
m=2/3
Equation of the line Slope-intercept form
y=mx+b
Put the value of any one point and slope in above equation
m=2/3, Point (-3,2)
2=(2/3)(-3)+b
2=-2+b
2+2=b
4=b
b=4
Now
m=2/3, Point (6,8)
8=(2/3)(6)+b
8=(2)(2)+b
8=4+b
8-4=b
4=b
b=4
b=4
Now again put the value of b and m in Equation of the Slope-intercept Form
y=mx+b
y=(2/3)x+4
Above equation is the Equation of the line passing through (-3,2) and (6,8)
Since the lines are paralallel their slops will be same so slope of the required line is 2/3 and it is passing through the point (-5,-4)
m=2/3, Point (-5,-4)
Put the above values in Equation of the line Slope-intercept form
y=mx+b
-4=(2/3)(-5)+b
-4=-10/3+b
-4+10/3=b
(-12+10)/3=b
-2/3=b
y-intercept of the requaired line b=-2/3
Put m=2/3 and b=-2/3 in the Equation of the line Slope-intercept form
y=mx+b
y=2/3(x)+(-2/3)
y=2x/3-2/3
Above equation is the requried equation of the line.
Graph
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Line passing through (-3, 2) and (6, 8)
y=(2/3)x+4
Line passing through (-5, -4)
y=2x/3-2/3
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