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Last updated date: 23rd Feb 2024
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IVSAT 2024
Answer
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Hint: We solve the given linear equation by simplifying the equation. We divide both sides of the equation $4x=24$ with 4. Then we simplify the right-hand side fraction to get the solution for $x$. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1.

Complete step by step answer:
The given equation $4x=24$ is a linear equation of $x$.
We divide both sides of the equation $4x=24$ with 4.
\[\begin{align}
  & \dfrac{4x}{4}=\dfrac{24}{4} \\
 & \Rightarrow x=\dfrac{24}{4} \\
\end{align}\]
We need to find the simplified form of the proper fraction \[\dfrac{24}{4}\].
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
This means we can’t eliminate any more common root from them other than 1.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our give fraction \[\dfrac{24}{4}\], the G.C.D of the denominator and the numerator is 4.
$\begin{align}
  & 2\left| \!{\underline {\,
  4,24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2,12 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,6 \,}} \right. \\
\end{align}$
The GCD is $2\times 2=4$.
Now we divide both the denominator and the numerator with 4 and get $\dfrac{{}^{24}/{}_{4}}{{}^{4}/{}_{4}}=\dfrac{6}{1}=6$.

Therefore, solving $4x=24$ we get 6.

Note: We also could have formed a factorisation of the equation $4x=24$. We take the constant 4 common out of the reformed equation $4x-24=0$.
Therefore, $4x-24=4\left( x-6 \right)=0$.
The multiplication of two terms gives 0 where one of the terms is positive and non-zero. This gives that the other term has to be zero.
So, $\left( x-6 \right)=0$ which gives $x=6$ as the solution.