In the rectangular coordinate system, the line 2y - 3x = 14 passes thr

*Method I (works for line equations that are relatively simple to graph with decent accuracy)

1. Rewrite equation in point-slope form, y = mx + b

2y - 3x = 14

2y = 3x + 14

y = \(\frac{3}{2}\)x + \(\frac{14}{2}\)

y = \(\frac{3}{2}\)x + 7

2. Find y- and x-intercepts

If x=0, y is 7

If y=0, x is \(\frac{- 14}{3}\)

The two intercept points are (0,7) and (-\(\frac{14}{3}\),0)

3. Graph line quickly. From graph, line passes through Q1, Q2, and Q3, and not Q4. Answer C

Method II

I used this one to be thorough. 57 seconds with double-checking. This method saves time with line equations whose numbers are hard to graph. To write it out makes it look complicated, but it isn't.

Step 1 from above: Rewrite equation in point-slope form, y = mx + b, y = \(\frac{3}{2}\)x + 7. Slope \(\frac{3}{2}\) is positive.

Step 2 from above: Find y- and x-intercepts. y = +7, x = - \(\frac{14}{3}\)

Step 3. Use properties of lines and quadrants. I memorized a few such properties by making a little time-saver chart for NEVER and ALWAYS and watching for a pattern and/or a mnemonic.

Step 3a: NEVER pattern: first N or P means slope is negative or positive, second N or P means y-intercept is negative or positive.

NN = never Q1 (i.e., if slope is negative (N) and y-intercept is negative (N), line never passes through Q1)
NP = never Q2
PN = never Q3
PP = never Q4

This pattern is easy for me to remember because of the way the letters, which correspond with Q1 to Q4 in order, fall out (double N, alphabetical order NP, switch order to PN, then double P)

Step 3b: ALWAYS pattern, don't laugh too hard at my mnemonic. Don't need ALWAYS pattern here unless you want to check other quadrants.

pOsitive slope has second letter O in it. O = Odd numbered quadrants
nEgative slope has second letter E in it. E = Even numbered quadrants

pOsitive slope ALWAYS passes through Q1 and Q3
nEgative slope ALWAYS passes through Q2 and Q4

Step 4: From line equation y = \(\frac{3}{2}\)x + 7, slope is positive, y-intercept is positive

Quickly, from chart above: PP = never Q4. Answer C


Optional Step 5: Ascertain by checking other quadrants

Slope of \(\frac{3}{2}\) is positive. Positive slope = always Q1 and Q3. Rule out answers A and D.

Q2? There's probably a rule for this context. I don't know it.

From Method I steps 2 and 3 above. Graph the line quickly with y-and x-intercepts of (0,7) and (-\(\frac{14}{3}\),0). Yes, it passes through Q2.

So line passes through Q1, Q2, and Q3. Rule out answers B and E. Only Answer C is left.

Hope it helps.


*I like pushpitkc 's method, was tempted to use it. But I'm not sure whether or not we can rule out Q4 completely by plugging in. Question pushpitkc or SajjadAhmad: Because y is a function of x, and any x choice must be positive to rule out Q4, what range of values for x do you choose and why? Is it enough to find one plugged-in value for x that doesn't satisfy?