Solve partial f partial a =cat+( a a-1 )y-z | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{\partial f}{\partial a}=cat+(\frac{a}{a-1})y-z$$

Solve for c

$\left\{\begin{matrix}c=-\frac{-ay+az-z}{at\left(1-a\right)}\text{, }&t\neq 0\text{ and }a\neq 1\text{ and }a\neq 0\\c\in \mathrm{R}\text{, }&\left(z=0\text{ and }a=0\right)\text{ or }\left(y=0\text{ and }z=0\text{ and }t=0\text{ and }a\neq 1\right)\text{ or }\left(a=\frac{z}{z-y}\text{ and }t=0\text{ and }y\neq 0\text{ and }y\neq z\right)\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $a-1$.

$$\left(a-1\right)\frac{\mathrm{d}}{\mathrm{d}a}(f)=cat\left(a-1\right)+ay-z\left(a-1\right)$$

Use the distributive property to multiply $a-1$ by $\frac{\mathrm{d}}{\mathrm{d}a}(f)$.

$$a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)=cat\left(a-1\right)+ay-z\left(a-1\right)$$

Use the distributive property to multiply $cat$ by $a-1$.

$$a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)=cta^{2}-cat+ay-z\left(a-1\right)$$

Use the distributive property to multiply $-z$ by $a-1$.

$$a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)=cta^{2}-cat+ay-za+z$$

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

$$cta^{2}-cat+ay-za+z=a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)$$

Subtract $ay$ from both sides.

$$cta^{2}-cat-za+z=a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)-ay$$

Add $za$ to both sides.

$$cta^{2}-cat+z=a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)-ay+za$$

Subtract $z$ from both sides.

$$cta^{2}-cat=a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)-ay+za-z$$

Combine all terms containing $c$.

$$\left(ta^{2}-at\right)c=a\frac{\mathrm{d}}{\mathrm{d}a}(f)-\frac{\mathrm{d}}{\mathrm{d}a}(f)-ay+za-z$$

The equation is in standard form.

$$\left(ta^{2}-at\right)c=-ay+az-z$$

Divide both sides by $ta^{2}-at$.

$$\frac{\left(ta^{2}-at\right)c}{ta^{2}-at}=\frac{-ay+az-z}{ta^{2}-at}$$

Dividing by $ta^{2}-at$ undoes the multiplication by $ta^{2}-at$.

$$c=\frac{-ay+az-z}{ta^{2}-at}$$

Divide $-ay+za-z$ by $ta^{2}-at$.

$$c=\frac{-ay+az-z}{at\left(a-1\right)}$$