Algebra Examples

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

Step 1

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

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

Step 2

Multiply both sides by $1 - r$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{a - r l}{1 - r} (1 - r)$ .

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Step 3.1.1.1

Cancel the common factor of $1 - r$ .

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Step 3.1.1.1.1

Cancel the common factor.

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

Step 3.1.1.1.2

Rewrite the expression.

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

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Move $r$ .

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

Step 3.1.1.2.2

Reorder $a$ and $- 1 l r$ .

$- 1 l r + a = S (1 - r)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $S (1 - r)$ .

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $S$ by $1$ .

$- 1 l r + a = S + S (- r)$

Step 3.2.1.2.2

Rewrite using the commutative property of multiplication.

$- 1 l r + a = S - S r$

Step 3.2.1.2.3

Move $S$ .

$- 1 l r + a = S - r S$

Step 3.2.1.2.4

Reorder $S$ and $- r S$ .

$- 1 l r + a = - r S + S$

Step 4

Solve for $l$ .

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Step 4.1

Rewrite $- 1 l$ as $- l$ .

$- l r + a = - r S + S$

Step 4.2

Subtract $a$ from both sides of the equation.

$- l r = - r S + S - a$

Step 4.3

Divide each term in $- l r = - r S + S - a$ by $- r$ and simplify.

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Step 4.3.1

Divide each term in $- l r = - r S + S - a$ by $- r$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.2.2

Cancel the common factor of $r$ .

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Step 4.3.2.2.1

Cancel the common factor.

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

Step 4.3.2.2.2

Divide $l$ by $1$ .

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Dividing two negative values results in a positive value.

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

Step 4.3.3.1.2

Cancel the common factor of $r$ .

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Step 4.3.3.1.2.1

Cancel the common factor.

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

Step 4.3.3.1.2.2

Divide $S$ by $1$ .

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

Step 4.3.3.1.3

Move the negative in front of the fraction.

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

Step 4.3.3.1.4

Dividing two negative values results in a positive value.

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