Question 1102492
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Here is another way -- that I find easier and faster -- to answer questions like this about lines parallel or perpendicular to a given line.<br>
(1) Put the equation in the form Ax+By=C.<br>
(2a) Every line parallel to the given line will have an equation also of the form Ax+By=K for some constant K.  (If K is the same constant as C, the lines are of course the same line, not parallel lines.)<br>
(2b) Every line perpendicular to the given line will have an equation of the form Bx-Ay=K for some constant K.  Note that the two coefficients have switched places, and one of them has changed sign.<br>
Let's apply this method to your problem, where we want an equation of the line through (3,7) perpendicular to the equation 3y = 4-2x.<br>
Step 1: put the equation in the required form:  2x+3y = 4.<br>
Step 2: Since we want a perpendicular line, it will have an equation of the form 3x-2y = K, where K is some constant.<br>
The constant is easily determined, knowing that the coordinates of the given point must satisfy the equation:
{{{3(3)-2(7) = 9-14 = -5}}}<br>
The equation (in this form) of the line we are looking for is 3x-2y = -5.