Answer

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**Hint:**Here, we will substitute different values of one variable in both the equations to find another variable. From there, we will get different coordinate points of both the lines. Then by using the coordinates of the lines, we will plot the graph for both the equations and mark the point of intersection of both the lines.

**Complete Step by Step Solution:**

We are given an equation of the line \[x + 3y + 3 = 0\] and \[3x - 2y + 6 = 0\].

\[ \Rightarrow x + 3y + 3 = 0\] …………………………………………………\[\left( 1 \right)\]

\[ \Rightarrow 3x - 2y + 6 = 0\]……………………………………………….\[\left( 2 \right)\]

Now, we will rewrite the equation \[\left( 1 \right)\], so we get

\[x = - 3 - 3y\] ………………………………………..\[\left( 3 \right)\]

By substituting \[y = 0\] in the above equation, we get

\[ \Rightarrow x = - 3 - 3\left( 0 \right) = - 3 - 0\]

\[ \Rightarrow x = - 3\]

Now, by substituting \[y = - 1\] in the equation \[\left( 3 \right)\], we get

\[x = - 3 - 3\left( { - 1} \right) = - 3 + 3\]

\[ \Rightarrow x = 0\]

Now, by substituting \[y = 1\] in the equation \[\left( 3 \right)\], we get

\[x = - 3 - 3\left( 1 \right) = - 3 - 3\]

Adding like terms, we get

\[ \Rightarrow x = - 6\]

So, the coordinates of the equation of line \[x + 3y + 3 = 0\] are \[\left( { - 3,0} \right)\],\[\left( {0, - 1} \right)\] and \[\left( { - 6,1} \right)\] .

Now, we will rewrite the equation \[\left( 2 \right)\], so we get

\[y = \dfrac{{3x + 6}}{2}\] ………………………………………\[\left( 4 \right)\]

Now, by substituting \[x = 0\] in the above equation, we get

\[ \Rightarrow y = \dfrac{{3\left( 0 \right) + 6}}{2} = \dfrac{6}{2}\]

\[ \Rightarrow y = 3\]

By substituting \[x = - 2\] in the equation \[\left( 4 \right)\], we get

\[y = \dfrac{{3\left( { - 2} \right) + 6}}{2} = \dfrac{{ - 6 + 6}}{2}\]

Simplifying the equation, we get

\[ \Rightarrow y = 0\]

Now, by substituting \[x = 2\] in the equation \[\left( 4 \right)\], we get

\[y = \dfrac{{3\left( 2 \right) + 6}}{2} = \dfrac{{6 + 6}}{2}\]

Simplifying the equation, we get

\[ \Rightarrow y = 6\]

So, the coordinates of the equation of line \[3x - 2y + 6 = 0\] are \[\left( {0,3} \right)\],\[\left( { - 2,0} \right)\] and \[\left( {2,6} \right)\] .

Now, we plot the graph for the coordinates and mark the intersection point, we get

**Note:**

We will solve the linear equation in two variables by the method of substitution to find the solution set at different points. Thus the solution set becomes the coordinates of the point in the graph for the linear equation. A linear equation in two variables is an equation with the highest power of 1 in two variables. Thus the graph of a linear equation is always a straight line. A point crossing the x-axis is called an x-intercept and A point crossing the y-axis is called the y-intercept. We can find the point of intersection by solving these two given equations but it is enough to mark the point of intersection only. Point of Intersection is the point where both the lines of the equations meet at a common point.

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