Solve for a pv(1+at)=qr(1+as)

$p v (1 + a t) = q r (1 + a s)$

Step 1

Simplify $p v (1 + a t)$ .

Rewrite.

$0 + 0 + p v (1 + a t) = q r (1 + a s)$

Simplify by adding zeros.

$p v (1 + a t) = q r (1 + a s)$

Apply the distributive property.

$p v \cdot 1 + p v (a t) = q r (1 + a s)$

Multiply $p$ by $1$ .

$p v + p v a t = q r (1 + a s)$

Step 2

Simplify $q r (1 + a s)$ .

Apply the distributive property.

$p v + p v a t = q r \cdot 1 + q r (a s)$

Multiply $q$ by $1$ .

$p v + p v a t = q r + q r a s$

Step 3

Subtract $q r a s$ from both sides of the equation.

$p v + p v a t - q r a s = q r$

Step 4

Subtract $p v$ from both sides of the equation.

$p v a t - q r a s = q r - p v$

Step 5

Factor $a$ out of $p v a t - q r a s$ .

Factor $a$ out of $p v a t$ .

$a (p v t) - q r a s = q r - p v$

Factor $a$ out of $- q r a s$ .

$a (p v t) + a (- q r s) = q r - p v$

Factor $a$ out of $a (p v t) + a (- q r s)$ .

$a (p v t - q r s) = q r - p v$

Step 6

Divide each term in $a (p v t - q r s) = q r - p v$ by $p v t - q r s$ and simplify.

Divide each term in $a (p v t - q r s) = q r - p v$ by $p v t - q r s$ .

$\frac{a (p v t - q r s)}{p v t - q r s} = \frac{q r}{p v t - q r s} + \frac{- p v}{p v t - q r s}$

Simplify the left side.

Cancel the common factor of $p v t - q r s$ .

Cancel the common factor.

$\frac{a (p v t - q r s)}{p v t - q r s} = \frac{q r}{p v t - q r s} + \frac{- p v}{p v t - q r s}$

Divide $a$ by $1$ .

$a = \frac{q r}{p v t - q r s} + \frac{- p v}{p v t - q r s}$

Simplify the right side.

Combine the numerators over the common denominator.

$a = \frac{q r - p v}{p v t - q r s}$