Given 12(x + 1) = 7(y − 1), write x in terms of y, and find y when x = 2.

Step-by-step solution

1. 1

   Expand both brackets first. Leaving the brackets in place makes it hard to see which terms move; expanding puts every term on a common footing:

   $12x+12=7y-7$

2. 2

   Move the constants to the side that keeps y. Subtract $12$ from both sides:

   $12x=7y-7-12=7y-19$

3. 3

   Divide to isolate x.

   $x=\frac{7y-19}{12}$

   The division applies to the whole numerator, which is why the bracket-free form $7y-19$ must be written over the single denominator $12$.

4. 4

   Substitute x = 2 into the original equation. Going back to the original avoids compounding any rearrangement error:

   $12(2+1)=7(y-1)\quad\Longrightarrow\quad 36=7y-7$

5. 5

   Solve for y.

   $7y=43\quad\Longrightarrow\quad y=\frac{43}{7}$

   $\boxed{x=\dfrac{7y-19}{12},\qquad y=\dfrac{43}{7}}$

6. 6

   Cross-check with the rearranged formula. Putting $y=\tfrac{43}{7}$ into $x=\tfrac{7y-19}{12}$ gives $x=\tfrac{43-19}{12}=\tfrac{24}{12}=2$ ✓, confirming that the rearrangement and the substitution agree.