Question

which graph matches this direct variation equation?
$y = \frac{2}{3}x$

Explanation:

Step1: Recall direct variation properties

Direct variation \( y = kx \) has a line through the origin (\( k \) is slope, positive here as \( k=\frac{2}{3}>0 \), so line rises from left to right).

Step2: Analyze slope \( \frac{2}{3} \)

Slope \( \frac{\text{rise}}{\text{run}}=\frac{2}{3} \). For \( x = 3 \), \( y=\frac{2}{3}(3)=2 \)? Wait, no: \( x = 3 \), \( y = 2 \)? Wait, no, \( y=\frac{2}{3}x \): when \( x = 3 \), \( y = 2 \); when \( x = -3 \), \( y=-2 \). Wait, no, let's check the graphs.

First graph: negative slope (falls left to right) – eliminate (since \( k>0 \)).

Second graph: let's check points. If \( x = 2 \), \( y=\frac{4}{3}\approx1.33 \), but the point shown is \( y = 3 \)? No, wait maybe I misread. Wait third graph: when \( x = 3 \), \( y = 2 \) (since \( \frac{2}{3}(3)=2 \)), and when \( x=-3 \), \( y=-2 \). Let's check the third graph: the line passes through (3,2) and (-3,-2), slope \( \frac{2 - (-2)}{3 - (-3)}=\frac{4}{6}=\frac{2}{3} \), and through origin. Second graph: if \( x = 2 \), \( y=\frac{4}{3}\approx1.33 \), but the point shown is \( y = 3 \) (maybe \( x = 2 \), \( y = 3 \)? Then slope would be \( \frac{3 - 0}{2 - 0}=\frac{3}{2} \), not \( \frac{2}{3} \)). Third graph: when \( x = 3 \), \( y = 2 \), slope \( \frac{2 - 0}{3 - 0}=\frac{2}{3} \), and passes through origin, with negative x (like x=-3, y=-2) which matches \( y=\frac{2}{3}x \) (since \( \frac{2}{3}(-3)=-2 \)).

Wait, let's re-express:

Direct variation \( y = \frac{2}{3}x \):

• Passes through (0,0).
• Slope \( \frac{2}{3} \) (rise 2, run 3).

So for \( x = 3 \), \( y = 2 \); \( x = -3 \), \( y = -2 \).

Third graph: check the points. The line goes through (3,2) and (-3,-2), and origin. Let's confirm:

From origin (0,0) to (3,2): rise 2, run 3, slope \( \frac{2}{3} \). Correct.

First graph: negative slope (slope \( \frac{-2 - 0}{3 - 0}=-\frac{2}{3} \)) – wrong.

Second graph: let's see, if \( x = 2 \), \( y = 3 \), slope \( \frac{3}{2} \) – wrong.

Third graph: slope \( \frac{2 - 0}{3 - 0}=\frac{2}{3} \), passes through origin, positive slope. So it's the third graph.

Answer:

The third graph (the right - most one) matches the equation \( y=\frac{2}{3}x \).