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Hint: We can find pairs by keeping the value of one variable in a given equation and another variable can easily be calculated as itâ€™s just a two-variable equation. Generally, sequential integers are to be used for our ease of understanding.

Complete step-by-step answer:

Here, we have to form such pairs using integers, be it positive or negative, such that $\dfrac{a}{b}=-3$. We can form these pairs by keeping integers on one variable sequentially or randomly, i.e.,

We can start substituting integral values in one variable, letâ€™s say for b

$\dfrac{a}{b}=-3$

Substituting b = 1 in above equation, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{1}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-3\]

One such pair formed by the values calculated above will be i.e., pair 1: (-3, 1).

Again, substituting sequential or random values for b in the given equation i.e., b = -3, 5, 6, 10, we get

For $b=-3$, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{-3}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[\begin{align}

&a=\left( -3 \right)\times \left( -3 \right) \\

&a=9 \\

\end{align}\]

Thus, pairs formed from similar operations in a given equation, we get pair 2: (9, -3).

Similarly, for$b=5$, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{5}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-15\]

Thus, pairs formed from similar operations in a given equation, we get pair 3: (-15, 5).

Now for \[b=6\], we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{6}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-18\]

Thus, pairs formed from similar operations in a given equation, we get pair 4: (-18, 6).

Now for \[b=10\], we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{10}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-30\]

Thus, pairs formed from similar operations in a given equation, we get pair 5: (-30, 10).

Hence, pairs formed using given conditionality, we get (-3 ,1); (9, -3); (-15, 5); (-18, 6); (-30, 10).

Note: Student might perform a mistake in the sign conventions while substituting values of either a or b in given equation like, \[\dfrac{6}{-2}=\dfrac{-6}{2}=-3\], hence pairs so formed could change from (6, -2) to (-6, 2), unknowingly.

Complete step-by-step answer:

Here, we have to form such pairs using integers, be it positive or negative, such that $\dfrac{a}{b}=-3$. We can form these pairs by keeping integers on one variable sequentially or randomly, i.e.,

We can start substituting integral values in one variable, letâ€™s say for b

$\dfrac{a}{b}=-3$

Substituting b = 1 in above equation, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{1}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-3\]

One such pair formed by the values calculated above will be i.e., pair 1: (-3, 1).

Again, substituting sequential or random values for b in the given equation i.e., b = -3, 5, 6, 10, we get

For $b=-3$, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{-3}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[\begin{align}

&a=\left( -3 \right)\times \left( -3 \right) \\

&a=9 \\

\end{align}\]

Thus, pairs formed from similar operations in a given equation, we get pair 2: (9, -3).

Similarly, for$b=5$, we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{5}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-15\]

Thus, pairs formed from similar operations in a given equation, we get pair 3: (-15, 5).

Now for \[b=6\], we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{6}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-18\]

Thus, pairs formed from similar operations in a given equation, we get pair 4: (-18, 6).

Now for \[b=10\], we get

\[\begin{align}

&\dfrac{a}{b}=-3 \\

&\dfrac{a}{10}=-3 \\

\end{align}\]

On cross-multiplying above equations, we get

\[a=-30\]

Thus, pairs formed from similar operations in a given equation, we get pair 5: (-30, 10).

Hence, pairs formed using given conditionality, we get (-3 ,1); (9, -3); (-15, 5); (-18, 6); (-30, 10).

Note: Student might perform a mistake in the sign conventions while substituting values of either a or b in given equation like, \[\dfrac{6}{-2}=\dfrac{-6}{2}=-3\], hence pairs so formed could change from (6, -2) to (-6, 2), unknowingly.

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