Simplify (7y - 3z) - 3(2y - z).

Step-by-step solution

1. 1

   Decide what has to happen before any combining. Like terms can only be collected once no parentheses remain, so the first job is distributing the $-3$ across $(2y-z)$. The leading parentheses have a coefficient of $+1$ and can simply be dropped.

2. 2

   Distribute the -3, tracking both signs. The multiplier is negative three, so each inner sign flips:

   $-3\cdot 2y=-6y,\qquad -3\cdot(-z)=+3z$

   Missing the second flip and writing $-3z$ is the single most common error in this problem. The expression becomes

   $7y-3z-6y+3z$

3. 3

   Group the like terms. Terms in $y$ go with terms in $y$, and $z$ with $z$ — they are not like terms with each other and can never be merged:

   $(7y-6y)+(-3z+3z)$

4. 4

   Combine each group.

   $7y-6y=y,\qquad -3z+3z=0$

   The $z$ terms annihilate completely, so the variable $z$ disappears from the answer.

5. 5

   State the result.

   $(7y-3z)-3(2y-z)=y$

   The expression is independent of $z$ — a surprisingly small answer for a two-variable start, which is worth double-checking.

6. 6

   Verify with random substitutions. Testing $50$ random pairs $(y,z)$ drawn from $[-9,9]$, the original expression equals $y$ every time to machine precision ✓. For instance at $y=4,z=7$: $(28-21)-3(8-7)=7-3=4=y$ ✓.