Solve for r s=(a-lr)/(1-r)

$s = \frac{a - l r}{1 - r}$

Step 1

Rewrite the equation as $\frac{a - l r}{1 - r} = s$ .

$\frac{a - l r}{1 - r} = s$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1 - r, 1$

Remove parentheses.

$1 - r, 1$

The LCM of one and any expression is the expression.

$1 - r$

Step 3

Multiply each term in $\frac{a - l r}{1 - r} = s$ by $1 - r$ to eliminate the fractions.

Multiply each term in $\frac{a - l r}{1 - r} = s$ by $1 - r$ .

$\frac{a - l r}{1 - r} (1 - r) = s (1 - r)$

Simplify the left side.

Cancel the common factor of $1 - r$ .

Cancel the common factor.

$\frac{a - l r}{1 - r} (1 - r) = s (1 - r)$

Rewrite the expression.

$a - l r = s (1 - r)$

Simplify the right side.

Apply the distributive property.

$a - l r = s \cdot 1 + s (- r)$

Simplify the expression.

Multiply $s$ by $1$ .

$a - l r = s + s (- r)$

Rewrite using the commutative property of multiplication.

$a - l r = s - s r$

Step 4

Solve the equation.

Add $s r$ to both sides of the equation.

$a - l r + s r = s$

Subtract $a$ from both sides of the equation.

$- l r + s r = s - a$

Factor $r$ out of $- l r + s r$ .

Factor $r$ out of $- l r$ .

$r (- l) + s r = s - a$

Factor $r$ out of $s r$ .

$r (- l) + r s = s - a$

Factor $r$ out of $r (- l) + r s$ .

$r (- l + s) = s - a$

Divide each term in $r (- l + s) = s - a$ by $- l + s$ and simplify.

Divide each term in $r (- l + s) = s - a$ by $- l + s$ .

$\frac{r (- l + s)}{- l + s} = \frac{s}{- l + s} + \frac{- a}{- l + s}$

Simplify the left side.

Cancel the common factor of $- l + s$ .

Cancel the common factor.

$\frac{r (- l + s)}{- l + s} = \frac{s}{- l + s} + \frac{- a}{- l + s}$

Divide $r$ by $1$ .

$r = \frac{s}{- l + s} + \frac{- a}{- l + s}$

Simplify the right side.

Move the negative in front of the fraction.

$r = \frac{s}{- l + s} - \frac{a}{- l + s}$

Combine the numerators over the common denominator.

$r = \frac{s - a}{- l + s}$

Factor $- 1$ out of $- l$ .

$r = \frac{s - a}{- (l) + s}$

Factor $- 1$ out of $s$ .

$r = \frac{s - a}{- (l) - 1 (- s)}$

Factor $- 1$ out of $- (l) - 1 (- s)$ .

$r = \frac{s - a}{- (l - s)}$

Rewrite negatives.

Rewrite $- (l - s)$ as $- 1 (l - s)$ .

$r = \frac{s - a}{- 1 (l - s)}$

Move the negative in front of the fraction.

$r = - \frac{s - a}{l - s}$

Solve for r s=(a-lr)/(1-r)

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