An equation to determine the value of l may be expressed in the form 5(k × l) = 2(k + l), where k is a constant. What is the value of k?

Steps to Solve

1. Distribute the constants on both sides of the equation Start with the given equation $5(k \times l) = 2(k + l)$. Distribute the constants on both sides to get rid of the parentheses. $$ 5kl = 2k + 2l $$

2. Isolate terms containing $l$ Move all terms containing $l$ to one side of the equation. $$ 5kl - 2l = 2k $$

3. Factor out $l$ Factor out $l$ from the left side of the equation. $$ l(5k - 2) = 2k $$

4. Solve for $l$ Divide both sides by $(5k - 2)$ to isolate $l$. $$ l = \frac{2k}{5k - 2} $$

5. Make $l$ the subject of the formula Since the problem states that the equation can be used to determine the value of $l$, let's rearrange the equation to make $k$ the subject. $$ l = \frac{2k}{5k - 2} $$ $$ l(5k - 2) = 2k $$ $$ 5kl - 2l = 2k $$ $$ 5kl - 2k = 2l $$ $$ k(5l - 2) = 2l $$ $$ k = \frac{2l}{5l-2} $$

6. Differentiate both sides with respect to $l$ Differentiate both sides of the equation $5(k \times l) = 2(k + l)$ with respect to $l$, remembering that $k$ is a constant.

$$ \frac{d}{dl} [5kl] = \frac{d}{dl} [2(k+l)] $$ $$ 5k = 2(0+1) $$ $$ 5k = 2 $$ $$ k = \frac{2}{5} $$