Question 849726
5x-4y=22 and -75x=150-60y


1. put both equations in y=mx+b form
2. find a point (x,y) on either equation
3. find equation of the perpendicular line that passes through (x,y)
4. use systematic equation to solve with equation 3 (the new one) and the one that wasn't used to get the point in step 2
5. calculate with distance formula


(The statement of your number 4 is not in the best wording).


BOTH EQUATIONS INTO SLOPE-INTERCEPT FORM
{{{y=(5/4)x-22/4}}}
{{{y=(5/4)x-11/2}}}----first equation
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{{{-25x=50-20y}}}
{{{-5x=10-4y}}}
{{{-5x-10=-4y}}}
{{{y=(5/4)x+10/4}}}
{{{y=(5/4)x+5/2}}}----second equation


PICK ANY POINT ON ONE EQUATION
y=(5/4)x+5/2
Let x=2.
y=(5/4)*2+5/2
y=5/2+5/2
y=5.
Point picked is (2, 5).


LINE PERPENDICULAR CONTAINING (2, 5):
Arbitrary choice to use point-slope formula.
Want slope {{{-4/5}}}.
{{{y-5=-(4/5)(x-2)}}}
{{{y-5=-(4/5)x+8/5}}}
{{{y=-(4/5)x+5+8/5}}}
{{{y=-(4/5)x+33/5}}}----Perpendicular to both of the given equations


INTERSECTION OF {{{y=(5/4)x-11/2}}} and {{{y=-(4/5)x+33/5}}} :
Obvious formulas for y are expected equal if the two equations intersect.
{{{(5/4)x-11/2=-(4/5)x+33/5}}}
Multiply members by 20 which is LCD.
{{{25x-110=-16x+33*4}}}
{{{25x-110=-16x+132}}}
(25+16)x=110+132=242
{{{41x=242}}}
{{{x=242/41}}}
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Find y.
{{{y=(5/4)(242/41)-11/2}}}
{{{y=(5*242)/(4*41)-(11/2)}}}
{{{y=(5*242)/(4*41)-(11/2)((2*41)/(2*41))}}}
{{{y=(5*242-11*2*41)/(4*41)}}}
{{{y=(5*121-11*41)/(82)}}
{{{y=966/82}}
{{{y=483/41}}}
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POINT on first equation is ( 242/41, 483/41 ).



FINDING DISTANCE BETWEEN THE TWO FOUND POINTS
Step 5 on your list.
You want to use the distance formula to determine or find what is the distance between ( 242/41, 483/41 )  and  (2, 5).  
Very possibly, if you would try to make a graph of the two given lines, you MIGHT find more convenient set of points to use, such as to pick a point on either line and possibly have a more convenient point on the other line, intersecting also a more convenient perpendicular to both.
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Distance is {{{highlight(sqrt((242/41-2)^2+(483/41-5)^2))}}}