Solve the system 220x + 100y = 960 and x + y = 6.

Step-by-step solution

1. 1

   Pick the easier equation to rearrange. The second equation has unit coefficients, so solving it for one variable costs nothing:

   $y=6-x$

   Choosing this direction avoids fractions entirely — rearranging the first equation instead would give $y=(960-220x)/100$.

2. 2

   Substitute into the first equation.

   $220x+100(6-x)=960$

   One equation in one unknown is now all that is left.

3. 3

   Expand and combine.

   $220x+600-100x=960\quad\Rightarrow\quad 120x+600=960$

   The coefficient $120=220-100$ is the difference of the two unit values, which is why a small change in $x$ moves the total so slowly.

4. 4

   Solve for $x$, then back-substitute.

   $120x=360\quad\Rightarrow\quad x=3,\qquad y=6-3=3$

5. 5

   Check both equations, and sanity-check by average. First: $220(3)+100(3)=660+300=960$ ✓. Second: $3+3=6$ ✓. As a structural check, the mean value per unit is $960/6=160$, exactly halfway between $220$ and $100$ — which forces an equal split, as found.