Chapter 4, Problem 1RE

Solve the following system by graphing

$\begin{array}{c}2x-y=4\\ x-2y=-4\end{array}$

To determine

To calculate: The solution of system of linear equations $2x-y=4$ and $x-2y=-4$ with the help of graphical method.

Answer to Problem 1RE

The solution of the equations $2x-y=4$ and $x-2y=-4$ are $x=4\text{ and }y=4$.

Explanation of Solution

Calculation:

Consider the provided equations.

$2x-y=4$ …… $\left(1\right)$

And

$x-2y=-4$..…. $\left(2\right)$

Draw the graph of equations by finding the respective intercepts.

For the equation $\left(1\right)$, $2x-y=4$

When $y=0$,

$\begin{array}{c}2x-0=4\\ 2x=4\\ x=2\end{array}$

When $x=0$,

$\begin{array}{c}2\cdot \left(0\right)-y=4\\ -y=4\\ y=-4\end{array}$

For the equation $\left(2\right)$, $x-2y=-4$

When $y=0$,

$\begin{array}{c}x-2\left(0\right)=-4\\ x=-4\end{array}$

When $x=0$,

$\begin{array}{c}0-2y=-4\\ -2y=-4\\ y=2\end{array}$

The graph of the equations bypassing through their respective intercept points is shown below,

From the above graph, the equations $\left(1\right)$ and $\left(2\right)$ intersect at a point, $\left(4,4\right)$. This implies that the point $\left(4,4\right)$ satisfies both the equations.

Thus, the solution of the provided system of equations is $x=4\text{ and }y=4$.

As a check, substitute the intersecting point coordinates $\left(4,4\right)$ in the equation $\left(1\right)$,

$\begin{array}{r}2x-y\stackrel{?}{=}4\\ 2\left(4\right)-4\stackrel{?}{=}4\\ 8-4\stackrel{?}{=}4\\ 4=4\end{array}$

Similarly, substitute $\left(4,4\right)$ in equation $\left(2\right)$,

$\begin{array}{r}x-2y\stackrel{?}{=}-4\\ 4-2\left(4\right)\stackrel{?}{=}-4\\ 4-8\stackrel{?}{=}-4\\ -4=-4\end{array}$

Hence, it is thus verified that $\left(4,4\right)$ satisfies the two equations

Therefore, solution of the linear equations is $x=4\text{ and }y=4$.