In the x-y coordinate plane, line k passes through the point (5, -4)

Reminder (to my students) / Suggestion (to the general reader):

1. One property at a time, till the end. Then the other. ("Don´t let your brain dance!")

2. Start with the easier checking property (or the one you feel more comfortable).
Indifferent? Start with the first.

\(?\,\,\left( 1 \right)\,\,\,\,:\,\,\,\,\left( {5, - 4} \right)\,\,\mathop \in \limits^? \,\,\,{\text{line}}\,\,\)

\(\left( i \right)\,\,\, - 4\,\,\mathop = \limits^? \,\, - \frac{2}{5}\left( 5 \right) - 2\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\)

\(\left( {ii} \right)\,\,\, - 4\,\,\mathop = \limits^? \,\, - \frac{6}{5}\left( 5 \right) + 2\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\)

\(\left( {iii} \right)\,\,\, - 4\,\,\mathop = \limits^? \,\, - \frac{7}{{10}}\left( 5 \right) - \frac{3}{2}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{Refute}}\,\,\left( C \right),\,\,\left( E \right)\)


\(?\,\,\left( 2 \right)\,\,\,\,:\,\,\,\,{\text{line}}\,\,x - {\text{intercept}}\,\,\,\mathop < \limits^? \,\,0\)

\(\left( i \right)\,\,0\, = \,\, - \frac{2}{5}\left( x \right) - 2\,\,\,\,\,\mathop \Rightarrow \limits^? \,\,\,x < \,\,0\,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{Refute}}\,\,\left( B \right)\)

\(\left( {ii} \right)\,\,0\, = \,\, - \frac{6}{5}\left( x \right) + 2\,\,\,\,\,\mathop \Rightarrow \limits^? \,\,\,x < \,\,0\,\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{Refute}}\,\,\left( D \right)\)


The correct answer is (A).


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Notice that, if the line passes through the origin, then the line has slope -4/5 (aka a slope of -0.8), but the line will not have a negative x-intercept
Notice that, if the line passes through the (0,4), then the line will have slope 0, BUT the line will not have a negative x-intercept. See below.


So, if the slope of the line is BETWEEN 0 and -0.8, then the line will have a negative x-intercept


Check the 3 statements:
i) y = -0.4x - 2
The slope = -0.4, which is BETWEEN 0 and -0.8. KEEP i for now

ii) y = 2 - 1.2x
Rewrite as: y = -1.2 + 2.
The slope = -1.2, which is NOT BETWEEN 0 and -0.8. So, ii cannot be true

iii) y = -0.7x - 1.5
The slope = -0.7, which is BETWEEN 0 and -0.8. KEEP iii for now

So far, i and iii COULD both be true.

However, in order for the point (5, -4) to be ON line k, its x and y coordinates must satisfy the equation of the line.
So, let's check i and iii

i) y = -0.4x - 2
Plug in x = 5 and y = -4 to get: -4 = -0.4(5) - 2 = -2 - 2 = -4
PERFECT!
So, i is true

iii) y = -0.7x - 1.5
Plug in x = 5 and y = -4 to get: -4 = -0.7(5) - 1.5 = -3.5 - 1.5 = -5
NO GOOD
So, iii is not true

Answer: A

Cheers,
Brent

We can at fist eliminate option II since if the line passes through its y intercept (substitute x=0 and obtain y=2) the line cannot pass also through a negative x intercept starting from point (5,-4)

Then we proceed by substituting the point on the equations to see if the line passes throigh the point and we obtain only I

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