Question

Which table of ordered pairs represents a proportional relationship

Answer 1

Last table is the answer

Explanation:

In this case, an proportional realtionship refer to the existence of a contant ratio of change between variables, that is, each y-value can be found by multiplying each x-value with this constant ratio of change.

So, notice that in the first table, is we multiply by four, you can get the first two pairs, but the last one doesn't fall into the ratio. That's not the answer.

Similarly, the second table doesn't have a constant ratio of change, because the last pair has different ratio.

However, the last table shows a constant ratio of change, because each x-value can be multiplied by -4, to get each y-value, that is

-3 x -4 = 12

-6 x -4 = 24

-9 x -4 = 36

Therefore, the right answer is the last table.

Answer 2

Last option

`\textbf{$\left[\begin{array}{cc}x & y\\-3 & 12\\-6 & 24\\-9 & 36\end{array}\right]$}`

Explanation:

Two variables have a proportional relationship if the ratios are equivalent. In other words, in this type of cases two quantities vary directly with each other, so we can write this in a mathematical language as follows:

`y=kx`

Here
`k` is the slope of the linear equation defined above. So, verifying that k is constant we have:

`k=(24-12)/(-6-(-3)))=(36-24)/(-9-(-6))=(36-12)/(-9-(-3))=-4 \\ \\ \therefore \boxed{k=-4}`

One way to prove this is by writing the equation that represents the table. From the two-point intercept form of the equation of a line we have:

`y-y_(1)=(y_(2)-y_(1))/(x_(2)-x_(1))(x-x_(1)) \\ \\ P(x_(1),y_(1))=P(-3,12) \\ \\ P(x_(2),y_(2))=P(-6,24) \\ \\ Subtituting \ x_(1), x_(2), y_(1), y_(2): \\ \\ y-12=(24-12)/(-6-(-3))(x-(-3)) \\ \\ y-12=-4(x+3) \\ \\ Solving: \\ \\ y-12=-4x-12 \\ \\ Adding \ 12 \ to \ both \ sides: \\ \\ y-12+12=-4x-12+12 \\ \\ \boxed{y=-4x}`

So, this implies that the ordered pairs of the last option represent a proportional relationship