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We know how to solve a linear equation in one variable. We shall now study a method for solving a system of two linear equations in two variables by transforming the two equations in two variables into one equation in one variable.
To make this transformation, we need to eliminate one equation and one variable. We can make this
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
$1,\text{\hspace{0.17em}}y=-3x+7.$
$\begin{array}{lll}y\hfill & =\hfill & -3\left(1\right)+7\hfill \\ y\hfill & =\hfill & -3+7\hfill \\ y\hfill & =\hfill & 4\hfill \end{array}$
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.
Slove the system $\{\begin{array}{l}5x-8y=18\\ 4x+\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}7\end{array}$
The point $\left(2,-1\right)$ is the point of intersection of the two lines.
The following rule alerts us to the fact that the two lines of a system are parallel.
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