Question

1. $\frac{1}{v} + \frac{3v + 12}{v^2 - 5v} = \frac{7v - 56}{v^2 - 5v}$

Explanation:

Step1: Find common denominator

The common denominator of the fractions is $v(v - 5)$ (since $v^2 - 5v = v(v - 5)$). Multiply each term by $v(v - 5)$ to eliminate denominators:
$$v(v - 5)\cdot\frac{1}{v}+v(v - 5)\cdot\frac{3v + 12}{v(v - 5)}=v(v - 5)\cdot\frac{7v - 56}{v(v - 5)}$$
Simplify each term:
$$(v - 5)+(3v + 12)=7v - 56$$

Step2: Simplify left side

Combine like terms on the left:
$$v - 5 + 3v + 12 = 4v + 7$$
So the equation becomes:
$$4v + 7 = 7v - 56$$

Step3: Solve for v

Subtract $4v$ from both sides:
$$7 = 3v - 56$$
Add 56 to both sides:
$$63 = 3v$$
Divide by 3:
$$v = 21$$

Step4: Check for restrictions

Original denominators: $v
eq0$ and $v^2 - 5v
eq0\implies v
eq0, v
eq5$. $v = 21$ is not a restriction, so it's valid.

Answer:

$v = 21$