Solve (a+b)x+a-b+1=x+6

$$( a + b ) x + a - b + 1 = x + 6$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{bx-x-b-5}{x+1}\text{, }&x\neq -1\\a\in \mathrm{C}\text{, }&x=-1\text{ and }b=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+bx+a-b+1=x+6$$

Subtract bx from both sides.

$$ax+a-b+1=x+6-bx$$

Add b to both sides.

$$ax+a+1=x+6-bx+b$$

Subtract 1 from both sides.

$$ax+a=x+6-bx+b-1$$

Subtract 1 from 6 to get 5.

$$ax+a=x+5-bx+b$$

Combine all terms containing a.

$$\left(x+1\right)a=x+5-bx+b$$

The equation is in standard form.

$$\left(x+1\right)a=5+b+x-bx$$

Divide both sides by x+1.

$$\frac{\left(x+1\right)a}{x+1}=\frac{5+b+x-bx}{x+1}$$

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

$$a=\frac{5+b+x-bx}{x+1}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax-x+a-5}{x-1}\text{, }&x\neq 1\\b\in \mathrm{C}\text{, }&x=1\text{ and }a=3\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+bx+a-b+1=x+6$$

Subtract ax from both sides.

$$bx+a-b+1=x+6-ax$$

Subtract a from both sides.

$$bx-b+1=x+6-ax-a$$

Subtract 1 from both sides.

$$bx-b=x+6-ax-a-1$$

Subtract 1 from 6 to get 5.

$$bx-b=x+5-ax-a$$

Combine all terms containing b.

$$\left(x-1\right)b=x+5-ax-a$$

The equation is in standard form.

$$\left(x-1\right)b=5-a+x-ax$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{5-a+x-ax}{x-1}$$

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

$$b=\frac{5-a+x-ax}{x-1}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx-x-b-5}{x+1}\text{, }&x\neq -1\\a\in \mathrm{R}\text{, }&x=-1\text{ and }b=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+bx+a-b+1=x+6$$

Subtract bx from both sides.

$$ax+a-b+1=x+6-bx$$

Add b to both sides.

$$ax+a+1=x+6-bx+b$$

Subtract 1 from both sides.

$$ax+a=x+6-bx+b-1$$

Subtract 1 from 6 to get 5.

$$ax+a=x+5-bx+b$$

Combine all terms containing a.

$$\left(x+1\right)a=x+5-bx+b$$

The equation is in standard form.

$$\left(x+1\right)a=5+b+x-bx$$

Divide both sides by x+1.

$$\frac{\left(x+1\right)a}{x+1}=\frac{5+b+x-bx}{x+1}$$

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

$$a=\frac{5+b+x-bx}{x+1}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax-x+a-5}{x-1}\text{, }&x\neq 1\\b\in \mathrm{R}\text{, }&x=1\text{ and }a=3\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+bx+a-b+1=x+6$$

Subtract ax from both sides.

$$bx+a-b+1=x+6-ax$$

Subtract a from both sides.

$$bx-b+1=x+6-ax-a$$

Subtract 1 from both sides.

$$bx-b=x+6-ax-a-1$$

Subtract 1 from 6 to get 5.

$$bx-b=x+5-ax-a$$

Combine all terms containing b.

$$\left(x-1\right)b=x+5-ax-a$$

The equation is in standard form.

$$\left(x-1\right)b=5-a+x-ax$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)b}{x-1}=\frac{5-a+x-ax}{x-1}$$

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

$$b=\frac{5-a+x-ax}{x-1}$$