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

Last updated date: 13th Jul 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.