Question

Solve the system of equation \[2x-y=2\] and $2x-3y=-6$ using a graphical method.

Answer 1

Hint: We start solving the problem by finding the points of intersection of the line \[2x-y=2\] with x-axis and y-axis by using the properties that the value of x-coordinate on y-axis is 0 and the value of y-coordinate on the x-axis is 0. We then find the points of intersection of the line $2x-3y=-6$ x-axis and y-axis. We then plot these lines in the graphs by joining the two obtained points. We then find the point of intersection of the lines from the plot which is the solution of the given system of equations.

Complete step-by-step answer:
According to the problem, we need to solve the given system of equations \[2x-y=2\] and $2x-3y=-6$.
Let us first find the points at which the line \[2x-y=2\] meets x-axis and y-axis.
We know that the value of x-coordinate on y-axis is 0.
So, we have $2\left( 0 \right)-y=2$.
$\Rightarrow -y=2$.
$\Rightarrow y=-2$.
So, the line \[2x-y=2\] touches the y-axis at the point $A\left( 0,-2 \right)$.
We also know that the value of y-coordinate on x-axis is 0.
So, we have $2x-0=2$.
$\Rightarrow 2x=2$.
$\Rightarrow x=1$.
So, the line \[2x-y=2\] touches the x-axis at the point $B\left( 1,0 \right)$.
Now, let us find the points at which the line \[2x-3y=-6\] meets x-axis and y-axis.
We know that the value of x-coordinate on y-axis is 0.
So, we have $2\left( 0 \right)-3y=-6$.
$\Rightarrow -3y=-6$.
$\Rightarrow y=2$.
So, the line \[2x-3y=-6\] touches the y-axis at the point $C\left( 0,2 \right)$.
We also know that the value of y-coordinate on x-axis is 0.
So, we have $2x-3\left( 0 \right)=-6$.
$\Rightarrow 2x=-6$.
$\Rightarrow x=-3$.
So, the line \[2x-3y=-6\] touches the x-axis at the point $D\left( -3,0 \right)$.
Let us plot these lines connecting the obtained points to get the intersection point of them.

From the plot, we can see that the point of intersection of the lines \[2x-y=2\] and $2x-3y=-6$ is $E\left( 3,4 \right)$.
So, the solution for the system of equations \[2x-y=2\] and $2x-3y=-6$ is $x=3$ and $y=4$.

Note: Whenever we get this type of problems, we first solve for the points of intersection with both axes as this will help us to draw the line accurately. We can also solve this problem analytically by substituting the equation of y obtained from first line in the equation of second line which will also give us the same result. We should see whether the given system has inequalities present in it before solving the problem. Similarly, we can expect problems to find the solution for the system of linear equations with more than two variables.