Question

Find the values of $x_1$ and $x_2$ where the following two constraints intersect.

$9x_1+7x_2\ge 57$
$4x_1+6x_2\ge 13$

Answer 1

Find the values of ${\mathit{x}}_{\mathbf{1}}$and ${\mathit{x}}_{\mathbf{2}}$.

The given constraints are,

$9x_1+7x_2\ge 57$

$4x_1+6x_2\ge 13$

Step 1: Rewrite the equations.

Convert the inequalities into equalities:

$9x_1+7x_2=57$

$4x_1+6x_2=13$

Isolate the equations $x_1$:

$\begin{array}{rcl}9x_1& =& 57-7x_2\\ & \Rightarrow & x_1=\frac{1}{9}\left(57-7x_2\right)\end{array}$

Step 2: Substitute $x_1$in the second equation.

$\begin{array}{rcl}4x_1+6x_2& =& 13\\ & \Rightarrow & 4\left(\frac{1}{9}\left(57-7x_2\right)\right)+6x_2=13\\ & \Rightarrow & \frac{4}{9}\left(57\right)-\frac{28x_2}{9}+6x_2=13\\ & \Rightarrow & \frac{26x_2}{9}=13-\frac{228}{9}\\ & \Rightarrow & \frac{26x_2}{9}=-\frac{37}{3}\\ & \Rightarrow & \frac{26x_2}{3}=-37\\ & \Rightarrow & 26x_2=-37\left(3\right)\\ & \Rightarrow & 26x_2=-111\\ & \Rightarrow & x_2=-\frac{111}{26}\\ & \Rightarrow & x_2=-4.269\end{array}$

Step 3: Substitute $\begin{array}{rcl}\mathbf{}{\mathit{x}}_{\mathbf{2}}& \mathbf{=}& \mathbf{-}\frac{\mathbf{111}}{\mathbf{26}}\end{array}$in the first equation.

$\begin{array}{rcl}9x_1+7x_2& =& 57\\ & \Rightarrow & 9x_1+7\left(-\frac{111}{26}\right)=57\\ & \Rightarrow & 9x_1=57+\frac{777}{26}\\ & \Rightarrow & x_1=\frac{1}{9}\left(57+\frac{777}{26}\right)\\ & \Rightarrow & x_1=\frac{1}{9}\left(\frac{2259}{26}\right)\\ & \Rightarrow & x_1=9.654\end{array}$

Therefore, the required values are $\begin{array}{rcl}{\mathit{x}}_{\mathbf{1}}& \mathbf{=}& \mathbf{9}\mathbf{.}\mathbf{654}\mathbf{,}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{269}\end{array}$.