First thing you have to do if you're going to write this line's equation in point-slope form is to find the slope. Use the points in the slope formula as follows: [tex] m=\frac{2-(-2)}{6-3}=\frac{4}{3} [/tex]. So the slope is 4/3. The point-slope form of a line is [tex] y-y_{1}=m(x-x_{1}) [/tex] where x1 is the x-coordinate from either point, and y1 is the y coordinate from the SAME point as the x. I'll use (3, -2) since that's the point they used in the solutions above. x is 3 and y is -2. Filling in accordingly, we have [tex] y+2=\frac{4}{3}(x-3) [/tex]. That is point-slope form. Now they want us to rewrite in standard form which is Ax + By = C, no fractions allowed and the x and the y have to be on the same side of the equals sign. However, before we can do that successfully, we need to rewrite the point-slope into slope-intercept. THEN we can do standard. In slope intercept, we will set the equation equal to y. [tex] y=\frac{4}{3}x-\frac{12}{3}-2 [/tex]. Simplifying a bit we get [tex] y = \frac{4}{3}x-4-2 [/tex] or [tex] y=\frac{4}{3}x-6 [/tex]. Now in order to get rid of the fraction we will multiply everything, every term, by 3. Doing that gives us [tex] 3y=4x-18 [/tex]. Getting x and y on the same side is [tex] -4x+3y=-18 [/tex]. I am not seeing this exact combination of point-slope and standard within the same answer. I checked my work several times, perhaps you can check how you posted the possible solutions?
