SOLUTION: write the equation of the line L. satisfying each of the following sets of geometric conditions. L passes through (5,6) and is perpendicular to 3x-5y=15 L has y-intercept(0,-3)

Question 93842: write the equation of the line L. satisfying each of the following sets of geometric conditions. L passes through (5,6) and is perpendicular to 3x-5y=15
L has y-intercept(0,-3) and is paralell to -3x+5y=-15
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
"L passes through (5,6) and is perpendicular to 3x-5y=15"

First convert 3x-5y=15 into slope intercept form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)

Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)

Start with the given equation

Subtract 3x from both sides

Simplify

Divide both sides by -5 to isolate y

Break up the fraction on the right hand side

Reduce and simplify

The original equation (standard form) is equivalent to (slope-intercept form)

The equation is in the form where is the slope and is the y intercept.

Now lets find the perpendicular line

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line

Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of , you can find the perpendicular slope by this formula:

where is the perpendicular slope

So plug in the given slope to find the perpendicular slope

When you divide fractions, you multiply the first fraction (which is really ) by the reciprocal of the second

Multiply the fractions.

So the perpendicular slope is

So now we know the slope of the unknown line is (its the negative reciprocal of from the line ). Also since the unknown line goes through (5,6), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

where m is the slope and (,) is the given point

Plug in , , and

Distribute

Multiply

Add to both sides to isolate y

Make into equivalent fractions with equal denominators

Combine the fractions

Reduce any fractions

So the equation of the line that is perpendicular to and goes through (,) is

So here are the graphs of the equations and

graph of the given equation (red) and graph of the line (green) that is perpendicular to the given graph and goes through (,)

"L has y-intercept(0,-3) and is paralell to -3x+5y=-15"

First convert -3x+5y=-15 into slope intercept form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)

Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)

Start with the given equation

Add 3x to both sides

Simplify

Divide both sides by 5 to isolate y

Break up the fraction on the right hand side

Reduce and simplify

The original equation (standard form) is equivalent to (slope-intercept form)

The equation is in the form where is the slope and is the y intercept.

Now lets find the parallel line

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line

The equation goes through the point (,) (which is equal and parallel to the given equation)

So the equation parallel to and that goes through (0,-3) is (or )
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