Given 9 + a - b = 5, find the value of a - b.

Step-by-step solution

1. 1

   Recognise what can and cannot be found. There is one equation but two unknowns, so $a$ and $b$ cannot be determined individually — infinitely many pairs work. What is pinned down is the single combination $a-b$, and that is exactly what the question asks for.

2. 2

   Group the terms you want together. The left side already contains $a-b$ as a block:

   $9+(a-b)=5$

   Seeing $a-b$ as one object, rather than two separate variables, turns the problem into a one-step equation.

3. 3

   Subtract 9 from both sides.

   $a-b=5-9=-4$

   The result is negative because $5$ is smaller than $9$: whatever $a$ and $b$ are, $a$ falls short of $b$ by $4$.

4. 4

   Verify with concrete pairs. Take $a=0$, $b=4$: then $9+0-4=5$ ✓ and $a-b=-4$ ✓. Take $a=10$, $b=14$: then $9+10-14=5$ ✓ and $a-b=-4$ ✓. Different individual values, the same difference — confirming that only the combination is determined.

5. 5

   State the family of solutions. Every solution has the form

   $a=b-4,\qquad b\ \text{arbitrary}$

   Geometrically this is a straight line of slope $1$ in the $(b,a)$ plane. To fix $a$ and $b$ separately, a second independent equation would be required.