SOLUTION: find the equation of the line which passes through the point of intersection of 5x-3y+2=0 and x+3y+1=0 and is parallel to 2x+3y+3=0

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Question 824664: find the equation of the line which passes through the point of intersection of 5x-3y+2=0 and x+3y+1=0 and is parallel to 2x+3y+3=0
Found 2 solutions by TimothyLamb, josmiceli:
Answer by TimothyLamb(4379) About Me  (Show Source):
You can put this solution on YOUR website!
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5x-3y+2=0
x+3y+1=0
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put the system of linear equations into standard form
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5x - 3y = -2
x + 3y = -1
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copy and paste the above standard form linear equations in to this solver:
https://sooeet.com/math/system-of-linear-equations-solver.php
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point of intersection:
x= -0.5 = -1/2
y= -0.1666666 = -1/6
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(given) our new line is parallel to:
2x + 3y + 3 = 0
in slope intercept form:
3y = -2x - 3
y = (-2/3)x - 1
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slope of our new parallel line = -2/3 = -0.666666
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to solve the linear equation, copy and paste this (the point of intersection and the parallel slope):
-0.5, -0.1666666, -0.666666
into the "Point-Slope form: x1 y1 m" input box here: https://sooeet.com/math/linear-equation-solver.php
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x-intercept = (-0.75, 0)
y-intercept = (0, -0.5)
slope = -0.666666
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answer:
the slope-intercept form of our new line:
y = (-0.666666)x - 0.5
y = (-2/3)x - 1/2
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Solve and graph linear equations:
https://sooeet.com/math/linear-equation-solver.php
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Solve quadratic equations, quadratic formula:
https://sooeet.com/math/quadratic-formula-solver.php
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Solve systems of linear equations up to 6-equations 6-variables:
https://sooeet.com/math/system-of-linear-equations-solver.php

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
First find the intersection:
Add the equations:
+5x+-+3y+%2B+2+=+0+
+x+%2B+3y+%2B+1+=+0+
+6x+%2B3+=+0+
+6x+=+3+
+x+=+1%2F2+
and
+x+%2B+3y+%2B+1+=+0+
+1%2F2+%2B+3y+%2B+1+=+0+
+3y+=+-3%2F2+
+y+=+-1%2F2+
The intersection is (1/2, -1/2 )
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Now find the slope of +2x+%2B+3y+%2B+3+=+0+
+3y+=+-2x+-+3+
+y+=+-%282%2F3%29%2Ax+-+1+
The slope is +m+=+-2%2F3+
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I can write the general point-slope equation:
+%28+y+-+%28-1%2F2%29+%29+%2F+%28+x+-+1%2F2+%29+=+-2%2F3+
+y+%2B+1%2F2+=+-2%2F3%2A%28+x+-+1%2F2+%29+
+y+%2B+1%2F2+=+%28-2%2F3%29%2Ax+%2B+1%2F3+
Multiply both sides by +6+
+6y+%2B+3+=+-4x+%2B+2+
+6y+=+-4x+-1+
+y+=+%28-2%2F3%29%2Ax+-+1%2F6+ answer
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check:
does it go through (1/2, -1/2) ?
+y+=+%28-2%2F3%29%2Ax+-+1%2F6+
+-1%2F2+=+%28-2%2F3%29%2A%281%2F2%29+-+1%2F6+
+-1%2F2+=+-1%2F3+-+1%2F6+
+-3%2F6+=+-2%2F6+-+1%2F6+
+-3%2F6+=+-3%2F6+
OK