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Question 73718: A line perpendicular to the line 4x-3y=7. It also has the same y-intercept as the line 5x-4y=20. What is the equation of the line?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! First let's find the slope of the graph of this equation:
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4x - 3y = 7
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Convert this line to the slope intercept form. We need to find its slope and the slope intercept
form makes this easy. The slope intercept form is of the form y = m*x + b where m, the
multiplier of the x term is the slope. So let's rearrange the given equation into this form.
Begin with the equation:
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4x - 3y = 7
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Eliminate the 4x on the left side by subtracting 4x from both sides to get:
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-3y = -4x + 7
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Now multiply both sides of this equation by -1 to get +3y on the left side:
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3y = 4x - 7
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Now solve for y by dividing both sides of this equation by 3. The result is:
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y = (4/3)x -(7/3)
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This is in the slope intercept form. The slope is the multiplier of the x. It is the
fraction (4/3). A line that is perpendicular to this line will have the negative inverse
of 4/3 as its slope. What is meant by negative inverse? First invert the fraction
so that 4/3 becomes 3/4. Then put a negative sign on it. So the slope of the perpendicular
line is -(3/4).
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So far so good. In slope intercept form the perpendicular line will be:
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y = -(3/4)x + b
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b is the value on the y-axis where the graph of this perpendicular line crosses the y-axis.
The problem says this intercept will be the same point as the line 5x - 4y = 20. So what we need
to do is find the value of y in this equation when x = 0. Think about it. Any point having
an x value of zero will be on the y-axis. So set x equal to zero and you get:
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5*0 - 4y = 20 which simplifies to:
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-4y = 20
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Find the value of the y intercept by dividing both sides by -4 to get y = -5. Now all
we have to do is return to our equation for the perpendicular line and put -5 in for b.
The equation is:
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y = (-3/4)*x + b and with the substitution this becomes:
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y = (-3/4)*x - 5
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That's it ... a line perpendicular to the line 4x - 3y = 7 and having the same y-intercept
as the line 5x - 4y = 20
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Hope this helps you to understand perpendicular line generation.
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