Solve the system x - y = 5 and 2x + y = 4.

Step-by-step solution

1. 1

   Look at the coefficients before choosing a method. The $y$ terms are $-y$ and $+y$ — already exact opposites. That makes elimination by addition the fastest route, with no multiplication needed to line them up.

2. 2

   Add the two equations. Adding left sides and right sides:

   $(x-y)+(2x+y)=5+4$

   $3x=9$

   The $y$ terms cancel because $-y+y=0$.

3. 3

   Solve for $x$.

   $x=\frac{9}{3}=3$

4. 4

   Back-substitute into the simpler equation. Using $x-y=5$ with $x=3$:

   $3-y=5\quad\Longrightarrow\quad -y=2\quad\Longrightarrow\quad y=-2$

5. 5

   Check the pair in BOTH original equations. Substituting is only a real check if the equation not used for back-substitution is tested too:

   $3-(-2)=5\qquad\checkmark$
   $2(3)+(-2)=6-2=4\qquad\checkmark$

   Geometrically the two lines have different slopes ($1$ and $-2$), so they meet at exactly one point, $(3,-2)$.