Solve a(x+a)-x=a(a+1)+1

$$a ( x + a ) - x = a ( a + 1 ) + 1$$

Solve for a

$$a=\frac{x+1}{x-1}$$

$$x\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by x+a.

$$ax+a^{2}-x=a\left(a+1\right)+1$$

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

$$ax+a^{2}-x=a^{2}+a+1$$

Subtract a^{2} from both sides.

$$ax+a^{2}-x-a^{2}=a+1$$

Combine a^{2} and -a^{2} to get 0.

$$ax-x=a+1$$

Subtract a from both sides.

$$ax-x-a=1$$

Add x to both sides.

$$ax-a=1+x$$

Combine all terms containing a.

$$\left(x-1\right)a=1+x$$

The equation is in standard form.

$$\left(x-1\right)a=x+1$$

Divide both sides by x-1.

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

Dividing by x-1 undoes the multiplication by x-1.

$$a=\frac{x+1}{x-1}$$

Solve for x

$$x=\frac{a+1}{a-1}$$

$$a\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by x+a.

$$ax+a^{2}-x=a\left(a+1\right)+1$$

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

$$ax+a^{2}-x=a^{2}+a+1$$

Subtract a^{2} from both sides.

$$ax-x=a^{2}+a+1-a^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$ax-x=a+1$$

Combine all terms containing x.

$$\left(a-1\right)x=a+1$$

Divide both sides by -1+a.

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

Dividing by -1+a undoes the multiplication by -1+a.

$$x=\frac{a+1}{a-1}$$