Solve 1/tleft(1+tright)=a/t+b/1+t

$$\frac{ 1 }{ t \left( 1+t \right) } = \frac{ a }{ t } + \frac{ b }{ 1+t }$$

Solve for a

$$a=-\frac{bt-1}{t+1}$$

$$t\neq 0\text{ and }t\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by t\left(t+1\right), the least common multiple of t\left(1+t\right),t,1+t.

$$1=\left(t+1\right)a+tb$$

Use the distributive property to multiply t+1 by a.

$$1=ta+a+tb$$

Swap sides so that all variable terms are on the left hand side.

$$ta+a+tb=1$$

Subtract tb from both sides.

$$ta+a=1-tb$$

Combine all terms containing a.

$$\left(t+1\right)a=1-tb$$

The equation is in standard form.

$$\left(t+1\right)a=1-bt$$

Divide both sides by 1+t.

$$\frac{\left(t+1\right)a}{t+1}=\frac{1-bt}{t+1}$$

Dividing by 1+t undoes the multiplication by 1+t.

$$a=\frac{1-bt}{t+1}$$

Solve for b

$$b=-\frac{at+a-1}{t}$$

$$t\neq 0\text{ and }t\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by t\left(t+1\right), the least common multiple of t\left(1+t\right),t,1+t.

$$1=\left(t+1\right)a+tb$$

Use the distributive property to multiply t+1 by a.

$$1=ta+a+tb$$

Swap sides so that all variable terms are on the left hand side.

$$ta+a+tb=1$$

Subtract ta from both sides.

$$a+tb=1-ta$$

Subtract a from both sides.

$$tb=1-ta-a$$

The equation is in standard form.

$$tb=1-a-at$$

Divide both sides by t.

$$\frac{tb}{t}=\frac{1-a-at}{t}$$

Dividing by t undoes the multiplication by t.

$$b=\frac{1-a-at}{t}$$