The relation y(b - x) = ax is non-linear. Change it into linear form Y = mX + c and state Y, X, m and c.

Step-by-step solution

1. 1

   Diagnose the type of non-linearity. Nothing here is an exponent - the trouble is the product $xy$ hidden in $y(b-x)=by-xy$. Taking logs would only make that worse. When variables multiply each other, the fix is usually reciprocals.

2. 2

   Divide by the product $xy$. Starting from $y(b-x)=ax$ and dividing both sides by $xy$ (valid for $x\neq 0$, $y\neq 0$):

   $\frac{b-x}{x}=\frac{a}{y}$

3. 3

   Split the left-hand fraction.

   $\frac{b}{x}-1=\frac{a}{y}$

   Both variables now appear only as $1/x$ and $1/y$ - that is the signal the transformation is working.

4. 4

   Solve for $1/y$. Divide through by $a$:

   $\frac{1}{y}=\frac{b}{a}\cdot\frac{1}{x}-\frac{1}{a}$

5. 5

   Match against $Y=mX+c$.

   $\underbrace{\frac{1}{y}}_{Y}=\underbrace{\frac{b}{a}}_{m}\underbrace{\frac{1}{x}}_{X}+\underbrace{\left(-\frac{1}{a}\right)}_{c}$

   So plot $1/y$ against $1/x$. The intercept gives $a=-1/c$ and then $b=ma$.

6. 6

   Check with numbers. Take $a=2,b=5,x=1$: the original gives $y(5-1)=2$, so $y=0.5$ and $1/y=2$. The linear form gives $\tfrac{5}{2}(1)-\tfrac12=2$ ✓. Note the intercept is negative, which is what tells you $a>0$.