Let $f(x) = \dfrac{2-3x}{5-2x}$. For what value of $a$ is $f(a) = 3$?
Thanks!
f = (2-3x)/(5-2x)
f(a) = 3 when
(2-3a)/(5-2a) = 3
2-3a = 15-6a
3a = 13
a = 13/3
To find the value of $a$ such that $f(a) = 3$, we need to solve the equation $f(a) = 3$.
Step 1: Start by substituting $f(a)$ with its expression $\dfrac{2-3a}{5-2a}$ in the equation:
$$\dfrac{2-3a}{5-2a} = 3$$
Step 2: Next, we can eliminate the fraction by multiplying both sides of the equation by $(5-2a)$:
$$(5-2a)\cdot \dfrac{2-3a}{5-2a} = 3 \cdot (5-2a)$$
Simplifying the left side, the $(5-2a)$ terms on both sides cancel out:
$$2-3a = 3 \cdot (5-2a)$$
Step 3: Distribute the $3$ on the right side:
$$2-3a = 15 - 6a$$
Step 4: Move the terms containing $a$ to one side of the equation by adding $6a$ to both sides:
$$2 - 3a + 6a = 15$$
$$2 + 3a = 15$$
Step 5: Move the constant term to the other side by subtracting $2$ from both sides:
$$3a = 15 - 2$$
$$3a = 13$$
Step 6: Finally, divide both sides of the equation by $3$ to solve for $a$:
$$a = \dfrac{13}{3}$$
Therefore, the value of $a$ that makes $f(a) = 3$ is $a = \dfrac{13}{3}$.
I hope this explanation was helpful! Let me know if you have any further questions.