Question

Give the geometric representations of \[2x + 9 = 0\] as an equation
(i)in one variable
(ii)in two variable

Answer 1

Hint: Here, we will first see that in one variable, there is only \[x\]-axis and no \[y\]-axis from the given equation and then we will simplify to find the value of \[x\] and then graph it on the axis. Then we will see in two variables, we have \[x\]-axis and \[y\]-axis from the given equation and then simplify the equation again to graph the equation.

Complete step-by-step answer:
We are given that the equation is
\[2x + 9 = 0{\text{ ......eq(1)}}\]
We know that in one variable, there is only \[x\]-axis and no \[y\]-axis from the given equation.
Now we will solve the equation (1) by subtracting both sides by 9, we get
\[
   \Rightarrow 2x + 9 - 9 = 0 - 9 \\
   \Rightarrow 2x = - 9 \\
 \]
Dividing the above equation by 2 on both sides, we get
\[
   \Rightarrow \dfrac{{2x}}{2} = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = - 4.5 \\
 \]
Graphing the above equation on the \[x\]–axis, we get

Here, the equation (1) is a point.
Hence, we can say that in one variable \[2x + 9 = 0\] is a point.

(ii)We know that in two variables, we have \[x\]-axis and \[y\]-axis from the given equation.
Hence, writing equation (1) in terms of \[x\] and \[y\], we get
\[
   \Rightarrow 2x + 9 = 0 \\
   \Rightarrow 0 + 2x + 9 = 0 \\
   \Rightarrow 0y + 2x + 9 = 0{\text{ ......eq.(2)}} \\
 \]
Putting \[y = 0\] in the equation (2), we get
\[
   \Rightarrow 0\left( 0 \right) + 2x + 9 = 0 \\
   \Rightarrow 0 + 2x + 9 = 0 \\
   \Rightarrow 2x + 9 = 0 \\
 \]

Subtracting the above equation on both sides by 9, we get
\[
   \Rightarrow 2x + 9 - 9 = 0 - 9 \\
   \Rightarrow 2x = - 9 \\
 \]
Dividing the above equation by 2 on both sides, we get
\[
   \Rightarrow \dfrac{{2x}}{2} = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = - 4.5 \\
 \]
So, the point is \[\left( { - 4.5,0} \right)\].
Putting \[y = 1\] in the equation (2), we get
\[
   \Rightarrow 0\left( 1 \right) + 2x + 9 = 0 \\
   \Rightarrow 0 + 2x + 9 = 0 \\
   \Rightarrow 2x + 9 = 0 \\
 \]
Subtracting the above equation on both sides by 9, we get
\[
   \Rightarrow 2x + 9 - 9 = 0 - 9 \\
   \Rightarrow 2x = - 9 \\
 \]
Dividing the above equation by 2 on both sides, we get
\[
   \Rightarrow \dfrac{{2x}}{2} = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = \dfrac{{ - 9}}{2} \\
   \Rightarrow x = - 4.5 \\
 \]
So, the point is \[\left( { - 4.5,1} \right)\].
Graphing the above points of the equation (2) on the \[x\]–axis and \[y\]-axis, we get

Here, the equation (2) is a line.
Hence, we can say that in two variables \[2x + 9 = 0\] is a line, which is parallel to the\[y\]–axis.

Note: While solving these types of questions, students should know that if you are asked to solve the equation with one or two variables, that would mean that we have to find some value of the variable like \[x\] from the given equation. This is a simple question: just graph the equation properly for one variable and two variables to avoid confusion and mistakes also.