Solve the system 0.3x − 0.4y = 2.2 and −0.2x − 0.5y = 1.6. Give the answer as an ordered pair.

Step-by-step solution

1. 1

   Clear the decimals first. Every coefficient has one decimal place, so multiplying both equations by $10$ produces an equivalent integer system and removes any risk of decimal arithmetic slips:

   $\begin{cases}3x-4y=22\\-2x-5y=16\end{cases}$

2. 2

   Choose a variable to eliminate. The $x$ coefficients are $3$ and $-2$, whose least common multiple is $6$. Multiply the first equation by $2$ and the second by $3$:

   $6x-8y=44,\qquad -6x-15y=48$

3. 3

   Add to eliminate x.

   $(6x-8y)+(-6x-15y)=44+48\quad\Longrightarrow\quad -23y=92\quad\Longrightarrow\quad y=-4$

4. 4

   Back-substitute to find x. Using $3x-4y=22$:

   $3x-4(-4)=22\quad\Longrightarrow\quad 3x+16=22\quad\Longrightarrow\quad 3x=6\quad\Longrightarrow\quad x=2$

   $\boxed{(2,\,-4)}$

5. 5

   Check in the original decimal equations. First: $0.3(2)-0.4(-4)=0.6+1.6=2.2$ ✓. Second: $-0.2(2)-0.5(-4)=-0.4+2.0=1.6$ ✓. Checking against the original decimals rather than the scaled integers also confirms the multiplication by $10$ was done correctly on every term.