Solve x(m+9)+y(m-2)=z(m-6)f

$$x ( m + 9 ) + y ( m - 2 ) = z ( m - 6 ) f$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{mx+9x+my-2y}{z\left(6-m\right)}\text{, }&m\neq 6\text{ and }z\neq 0\\f\in \mathrm{R}\text{, }&\left(m=6\text{ and }x=-\frac{4y}{15}\right)\text{ or }\left(m=-9\text{ and }z=0\text{ and }y=0\right)\text{ or }\left(x=-\frac{y\left(m-2\right)}{m+9}\text{ and }m\neq -9\text{ and }z=0\text{ and }m\neq 6\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by m+9.

$$xm+9x+y\left(m-2\right)=z\left(m-6\right)f$$

Use the distributive property to multiply y by m-2.

$$xm+9x+ym-2y=z\left(m-6\right)f$$

Use the distributive property to multiply z by m-6.

$$xm+9x+ym-2y=\left(zm-6z\right)f$$

Use the distributive property to multiply zm-6z by f.

$$xm+9x+ym-2y=zmf-6zf$$

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

$$zmf-6zf=xm+9x+ym-2y$$

Combine all terms containing f.

$$\left(zm-6z\right)f=xm+9x+ym-2y$$

The equation is in standard form.

$$\left(mz-6z\right)f=mx+9x+my-2y$$

Divide both sides by zm-6z.

$$\frac{\left(mz-6z\right)f}{mz-6z}=\frac{mx+9x+my-2y}{mz-6z}$$

Dividing by zm-6z undoes the multiplication by zm-6z.

$$f=\frac{mx+9x+my-2y}{mz-6z}$$

Divide xm+9x+ym-2y by zm-6z.

$$f=\frac{mx+9x+my-2y}{z\left(m-6\right)}$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{9x-2y+6fz}{x+y-fz}\text{, }&x\neq fz-y\\m\in \mathrm{R}\text{, }&x=-\frac{4fz}{11}\text{ and }y=\frac{15fz}{11}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by m+9.

$$xm+9x+y\left(m-2\right)=z\left(m-6\right)f$$

Use the distributive property to multiply y by m-2.

$$xm+9x+ym-2y=z\left(m-6\right)f$$

Use the distributive property to multiply z by m-6.

$$xm+9x+ym-2y=\left(zm-6z\right)f$$

Use the distributive property to multiply zm-6z by f.

$$xm+9x+ym-2y=zmf-6zf$$

Subtract zmf from both sides.

$$xm+9x+ym-2y-zmf=-6zf$$

Subtract 9x from both sides.

$$xm+ym-2y-zmf=-6zf-9x$$

Add 2y to both sides.

$$xm+ym-zmf=-6zf-9x+2y$$

Combine all terms containing m.

$$\left(x+y-zf\right)m=-6zf-9x+2y$$

The equation is in standard form.

$$\left(x+y-fz\right)m=-9x+2y-6fz$$

Divide both sides by x+y-fz.

$$\frac{\left(x+y-fz\right)m}{x+y-fz}=\frac{-9x+2y-6fz}{x+y-fz}$$

Dividing by x+y-fz undoes the multiplication by x+y-fz.

$$m=\frac{-9x+2y-6fz}{x+y-fz}$$