Question

How do you solve the system $2x - y = 5$ and $x = 4$ by graphing?

Answer 1

For the first equation, write it in slope intercept form and plot it on a graph. Mark the $y$-intercept first and substitute two values to roughly get the straight line. Now plot the second equation whose line will be parallel to $Y$-axis. Now find the intersection of both lines. The intersection point will be the solution to the equation.

Formula used:
Any straight line can be written in slope-intercept form, $y = mx + b$
where $m$ is the slope of the line,$m = \tan \theta$ and $b$ is the intercept.

Complete step-by-step answer:
Given the system of equations,
$2x - y = 5$,
$x = 4$.
First, let’s plot $2x - y = 5$.
Converting it into slope intercept form,
$y = 2x - 5$
where $m = 2;b = - 5$

In the above graph, we first plotted the $y$-intercept $(0, - 5)$
To get a rough view of the line we need two more coordinates.
So, we substitute $x = 0$ in the equation.
$\Rightarrow 2(0) - 5 = y$
$\Rightarrow y = - 5$
Now, we have another coordinate,$(0, - 5)$ which is the same as the $y$-intercept.
So now we substitute $y = 0$ in the equation.
$\Rightarrow 2x - 5 = 0$
$\Rightarrow x = \dfrac{5}{2}$
Now we have another coordinate $(\dfrac{5}{2},0)$. Plot this and join all the points to get a straight line.
Since we are done plotting the first equation, we shall now graph the second equation,$x = 4$
When we write this equation in the slope-intercept form we get, $m = \infty ;b = 0$
If $m = \infty$, That means $\tan \theta = \infty$which only happens if $\tan \theta = 90^\circ$
This is the reason our second equation is a straight line at $90^\circ$ parallel to $y$ axis.
After plotting $x = 4$, the graph will look like this.
$\therefore$From the graph, we can clearly see that the two lines intersect at the point,$(4,3)$.
We can cross-check by substituting in both the equations to know if our answer is right.
Substituting $(4,3)$in $2x - y = 5$,$x = 4;y = 3$
$\Rightarrow 2(4) - 3 = 5$
Upon opening the bracket we get,
$\Rightarrow$ $8 - 3 = 5$
$\Rightarrow 5 = 5$

Hence the coordinates satisfy both the equations.

Additional information: The only point which satisfies both the equations and is the solution of both the equations is the point of intersection. This graphing technique can be used for more than $2$ equations to easily find the intersection point or the common solution to the given equations.

Note:
After getting an answer, one must always cross-check by substituting the values back in the equation to see if they are correct. $2$ or $3$ values can be taken for substitution to get a rough sketch of the given straight line.