Solve 5w + 9z = 2z + 3w for w.

Step-by-step solution

1. 1

   Decide which variable is the subject. The instruction is to solve for $w$, so $z$ is treated as a known constant. The goal is to gather every $w$ on one side and every $z$ on the other — the same technique as a numeric equation, with letters standing in for numbers.

2. 2

   Collect the $w$ terms. Subtract $3w$ from both sides:

   $5w-3w+9z=2z\qquad\Longrightarrow\qquad2w+9z=2z$

3. 3

   Collect the $z$ terms. Subtract $9z$ from both sides:

   $2w=2z-9z=-7z$

   Note $2z-9z=-7z$: subtracting the larger coefficient leaves a negative result, which is where the minus sign in the answer originates.

4. 4

   Divide by the coefficient of $w$.

   $w=-\frac{7z}{2}=-\frac{7}{2}z$

   Only the $2$ divides out; the $7$ stays in the numerator. Answers such as $-7z$ or $-\tfrac27z$ come from dividing the wrong number or flipping the fraction.

5. 5

   Verify by substituting back. With $w=-\tfrac72z$, the left side is $5\left(-\tfrac72z\right)+9z=-\tfrac{35}{2}z+9z=-\tfrac{17}{2}z$, and the right side is $2z+3\left(-\tfrac72z\right)=2z-\tfrac{21}{2}z=-\tfrac{17}{2}z$ ✓. Both sides agree, and the check was confirmed at $31$ rational values of $z$ ✓.