Solve the system 2x + 3y = 7 and x - y = 1.

Step-by-step solution

1. 1

   Choose the equation that is cheapest to rearrange. The second equation, $x-y=1$, already has a coefficient of $1$ on $x$, so isolating a variable there introduces no fractions. That is the whole reason substitution beats elimination on this system.

2. 2

   Solve the easy equation for one variable. Add $y$ to both sides of $x-y=1$:

   $x=y+1$

3. 3

   Substitute that expression into the other equation. Replacing $x$ in $2x+3y=7$ turns a two-variable system into a single equation in $y$:

   $2(y+1)+3y=7$

   $2y+2+3y=7\quad\Longrightarrow\quad 5y+2=7$

4. 4

   Solve the resulting linear equation. Subtract $2$, then divide by $5$:

   $5y=5\quad\Longrightarrow\quad y=1$

5. 5

   Back-substitute to recover the other variable. Put $y=1$ into $x=y+1$:

   $x=1+1=2$

6. 6

   Check the pair in both original equations, not just one. $2(2)+3(1)=4+3=7$ and $2-1=1$. Both hold, so $(2,1)$ is the unique intersection point of the two lines.