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How do you solve for $L$ in $A = \dfrac{r}{{2L}}$?

Last updated date: 25th Feb 2024
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IVSAT 2024
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Hint:This question is related to linear equation concept. An equation for a straight line is known as a linear equation. The term which is involved in a linear equation is either a constant or a single variable or product of a constant. The two variables can never be multiplied. All linear equations have a line graph. Linear equations are the same as linear function. The general form of writing a linear equation is$y = mx + c$ and $m$ is not equal to zero, where $m$ is the slope and $c$ is the point on which it cuts the y-axis. $y = mx + c$ is also known as the equation of the line in slope-intercept form. This given question deals with a specific type of linear equation and that is, formulas for problem solving.

Complete step by step solution:
Given is $A = \dfrac{r}{{2L}}$
We have to solve the given equation in order to find the value of $L$ for which the left-hand side is equal to the right-hand side of the equation.
Let us simply start by simplifying the given equation by multiplying both sides of the equation by $2$.
\Rightarrow A = \dfrac{r}{{2L}} \\
\Rightarrow A \times 2 = \dfrac{r}{{2L}} \times 2 \\
\Rightarrow 2A = \dfrac{r}{L} \\
Next, let us multiply $L$ on both the sides of the equation and we get,
\Rightarrow 2A = \dfrac{r}{L} \\
\Rightarrow 2A \times L = \dfrac{r}{L} \times L \\
\Rightarrow 2AL = r \\
Now, we isolate $L$ on the left-hand side of the equation by dividing both the sides of the equation by $2A$ and we get,
\Rightarrow 2AL = r \\
\Rightarrow \dfrac{{2AL}}{{2A}} = \dfrac{r}{{2A}} \\
\Rightarrow L = \dfrac{r}{{2A}} \\
Therefore, the value of $L$ is $\dfrac{r}{{2A}}$.

Note: Now that we know the value of $L$ is $\dfrac{r}{{2A}}$, there is a way to double check our answer. In order to double check the solution we are supposed to substitute the value of $L$ which we got as
$\dfrac{r}{{2A}}$ in the given equation, $A = \dfrac{r}{{2L}}$
\Rightarrow A = \dfrac{r}{{2L}} \\
\Rightarrow A = \dfrac{r}{{2\left( {\dfrac{r}{{2A}}} \right)}} \\
\Rightarrow A = \dfrac{r}{2} \times \dfrac{{2A}}{r} \\
\Rightarrow A = A \\
Now, the left-hand side is equal to the right-hand side of the equation. So, we can conclude that our solution or the value of $L$ which we calculated was correct.
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