11.2 Elimination by substitution

Solve the system $\begin{array}{lll}\{\begin{array}{l}2x+3y=14\\ 3x+y=7\end{array}\hfill & \hfill & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\hfill \end{array}$

Step 1:  Since the coefficient of $y$ in equation 2 is 1, we will solve equation 2 for $y$ .

       $y=-3x+7$

Step 2:  Substitute the expression $-3x+7$ for $y$ in equation 1.

       $2x+3\left(-3x+7\right)=14$

Step 3:  Solve the equation obtained in step 2.
      $\begin{array}{rrr}\hfill 2x+3\left(-3x+7\right)& =\hfill & 14\hfill \\ \hfill 2x-9x+21& =\hfill & 14\hfill \\ \hfill -7x+21& =\hfill & 14\hfill \\ \hfill -7x& =\hfill & -7\hfill \\ \hfill x& =\hfill & 1\hfill \end{array}$
Step 4:  Substitute $x=1$ into the equation obtained in step $1,y=-3x+7.$
      $\begin{array}{lll}y\hfill & =\hfill & -3\left(1\right)+7\hfill \\ y\hfill & =\hfill & -3+7\hfill \\ y\hfill & =\hfill & 4\hfill \end{array}$
 We now have $x=1$ and $y=4.$

Step 5:  Substitute $x=1,y=4$ into each of the original equations for a check.
$$\begin{array}{llllllllllll}(1)\hfill & \hfill & \hfill 2x+3y& =\hfill & 14\hfill & \hfill & (2)\hfill & \hfill & \hfill 3x+y& =\hfill & 7\hfill & \hfill \\ \hfill & \hfill & \hfill 2(1)+3(4)& =\hfill & 14\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill 3(1)+(4)& =\hfill & 7\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 2+12& =\hfill & 14\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill 3+4& =\hfill & 7\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 14& =\hfill & 14\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill & \hfill & \hfill 7& =\hfill & 7\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$$

Step 6:  The solution is $\left(1,4\right).$ The point $\left(1,4\right)$ is the point of intersection of the two lines of the system.