SOLUTION: Please write an equation of the line through (-5,4) and perpendicular to the line 5x + 4y = 24. Thank You

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Question 146191: Please write an equation of the line through (-5,4) and perpendicular to the line 5x + 4y = 24. Thank You
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

5x+%2B+4y+=+24 Start with the given equation.


4y=24-5x Subtract 5x from both sides.


y=%2824-5x%29%2F%284%29 Divide both sides by 4 to isolate y.


y=-%285%2F4%29x%2B6 Rearrange the terms and simplify.


We can see that the equation y=-%285%2F4%29x%2B6 has a slope m=-5%2F4 and a y-intercept b=6.


Now to find the slope of the perpendicular line, simply flip the slope m=-5%2F4 to get m=-4%2F5. Now change the sign to get m=4%2F5. So the perpendicular slope is m=4%2F5.


Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope m=-5%2F4 and the coordinates of the given point .


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y-4=%284%2F5%29%28x--5%29 Plug in m=4%2F5, x%5B1%5D=-5, and y%5B1%5D=4


y-4=%284%2F5%29%28x%2B5%29 Rewrite x--5 as x%2B5


y-4=%284%2F5%29x%2B%284%2F5%29%285%29 Distribute


y-4=%284%2F5%29x%2B4 Multiply


y=%284%2F5%29x%2B4%2B4 Add 4 to both sides.


y=%284%2F5%29x%2B8 Combine like terms.


So the equation of the line perpendicular to 5x+%2B+4y+=+24 that goes through the point is y=%284%2F5%29x%2B8.


Here's a graph to visually verify our answer:
Graph of the original equation y=-%285%2F4%29x%2B6 (red) and the perpendicular line y=%284%2F5%29x%2B8 (green) through the point .