Question

which of the following tables represents a linear relationship that is also proportional? \
\

x	-3	0	3	\
y	0	2	4	\

\

x	-4	-2	0	\
y	-2	0	2	\

\

x	-4	0	4	\
y	1	0	-1	\

\

x	-1	0	1	\
y	-5	-3	-1

Explanation:

Step1: Recall proportional linear rules

A proportional linear relationship follows $y=kx$ (passes through origin, constant $\frac{y}{x}$).

Step2: Check Table 1

$\frac{0}{-3}=0$, $\frac{2}{0}$ undefined. Not proportional.

Step3: Check Table 2

$\frac{-2}{-4}=\frac{1}{2}$, $\frac{0}{-2}=0$, $\frac{2}{0}$ undefined. Not proportional.

Step4: Check Table 3

$\frac{1}{-4}=-\frac{1}{4}$, $\frac{0}{0}$ undefined, $\frac{-1}{4}=-\frac{1}{4}$. Fails origin check.

Step5: Recheck Table 3 correction

Wait, $\frac{y}{x}$ for valid pairs: $\frac{1}{-4}=-\frac{1}{4}$, $\frac{-1}{4}=-\frac{1}{4}$. And $(0,0)$ is included, so $y=-\frac{1}{4}x$, which is proportional.

Step6: Verify Table 4

$\frac{-5}{-1}=5$, $\frac{-3}{0}$ undefined, $\frac{-1}{1}=-1$. Ratios not equal. Not proportional.

Answer:

The table with $x: -4, 0, 4$ and $y: 1, 0, -1$ (third option)