How do you solve the proportion \[\dfrac{9}{6}=\dfrac{x}{8}\]?

Hint: To solve the given proportion, we first have to convert it to an equation. We should also know that to solve a linear equation, we can solve a linear equation by taking the variable to one side of the equation and the constants to another side. By this, we can find the value of the variable that satisfies the given equation.

Complete step by step answer:
We are given the proportion \[\dfrac{9}{6}=\dfrac{x}{8}\], to solve this proportion we need to convert it to an equation. We can do this as follows,
\[\Rightarrow \dfrac{9}{6}=\dfrac{x}{8}\]
Subtracting \[\dfrac{9}{6}\] from both sides, we get
\[\begin{align}
  & \Rightarrow \dfrac{9}{6}-\dfrac{9}{6}=\dfrac{x}{8}-\dfrac{9}{6} \\
 & \Rightarrow 0=\dfrac{x}{8}-\dfrac{9}{6} \\
\end{align}\]
Thus, we have converted the proportion to an equation. As the degree of the equation is one, it is a linear equation. Flipping the sides of the above equation, we get
\[\Rightarrow \dfrac{x}{8}-\dfrac{9}{6}=0\]
Multiplying both sides of the above equation by 8, we get
\[\begin{align}
  & \Rightarrow 8\left( \dfrac{x}{8}-\dfrac{9}{6} \right)=\left( 0 \right)8 \\
 & \Rightarrow 8\left( \dfrac{x}{8} \right)-8\left( \dfrac{9}{6} \right)=0 \\
 & \Rightarrow x-\dfrac{72}{6}=0 \\
\end{align}\]
Adding \[\dfrac{72}{6}\] to both sides, we get
\[\begin{align}
  & \Rightarrow x-\dfrac{72}{6}+\dfrac{72}{6}=0+\dfrac{72}{6} \\
 & \Rightarrow x=\dfrac{72}{6} \\
\end{align}\]
The GCF of the numerator and denominator is 6, dividing both by this, we get
\[\begin{align}
  & \Rightarrow x=\dfrac{\dfrac{72}{6}}{\dfrac{6}{6}}=\dfrac{12}{1} \\
 & \therefore x=12 \\
\end{align}\]
 Hence, the solution of the given proportion is \[x=12\].

Note:
We can also directly solve this proportion without converting to equation, as follows
\[\dfrac{9}{6}=\dfrac{x}{8}\]
Flipping the sides, we get
\[\Rightarrow \dfrac{x}{8}=\dfrac{9}{6}\]
Multiplying both sides by 8, we get
\[\begin{align}
  & \Rightarrow 8\left( \dfrac{x}{8} \right)=8\left( \dfrac{9}{6} \right) \\
 & \Rightarrow x=\dfrac{72}{6} \\
\end{align}\]
The GCF of the numerator and denominator is 6, dividing both by this, we get
\[\begin{align}
  & \Rightarrow x=\dfrac{\dfrac{72}{6}}{\dfrac{6}{6}}=\dfrac{12}{1} \\
 & \therefore x=12 \\
\end{align}\]
But we can do this only for this proportion as the degree of the variable is one, if the degree is larger than one. Then we have to convert it to an equation, and then solve it to find the value satisfying.