Question

The equation \[2x - y = 4\] meets
(a) \[x\]–axis at \[\left( { - 4,0} \right)\]
(b) \[y\]–axis at \[\left( {0,2} \right)\]
(c) \[x\]–axis at \[\left( {2,0} \right)\]
(d) \[y\]–axis at \[\left( {0, - 4} \right)\]

Answer 1

Hint: Here, we need to find which of the given options is correct. We will substitute the points given in the options in the given equation. We will find the points of intersection of the given equation, and both the axes to find the correct options.

Complete step by step solution:
First, we will find the points of intersection of the given line and the \[x\]–axis.
The \[x\]–axis is given by the equation \[y = 0\].
The value of the ordinate at each point on the \[x\]–axis is 0, irrespective of the value of the abscissa.
We need to find the point of intersection of the equations \[2x - y = 4\] and \[y = 0\].
Substituting \[y = 0\] in the equation \[2x - y = 4\], we get
\[ \Rightarrow 2x - 0 = 4\]
Subtracting the terms in the equation, we get
$\Rightarrow 2x=4$
Dividing both sides by 2, we get
\[ \Rightarrow x = 2\]
Thus, if \[y = 0\], then \[x = 2\].
Therefore, the point of intersection of the line \[2x - y = 4\] and the equation \[y = 0\] is \[\left( {2,0} \right)\].
Hence, the equation \[2x - y = 4\] meets the \[x\]–axis at \[\left( {2,0} \right)\].
Thus, option (c) is one of the correct options.
Next, we will find the points of intersection of the given line and the \[y\]–axis.
The \[y\]–axis is given by the equation \[x = 0\].
The value of the abscissa at each point on the \[y\]–axis is 0, irrespective of the value of the ordinate.
We need to find the point of intersection of the equations \[2x - y = 4\] and \[x = 0\].
Substituting \[x = 0\] in the equation \[2x - y = 4\], we get
\[ \Rightarrow 2\left( 0 \right) - y = 4\]
Multiplying the terms in the equation, we get
\[ \Rightarrow 0 - y = 4\]
Subtracting the terms in the equation, we get
\[ \Rightarrow - y = 4\]
Multiplying both sides by \[ - 1\], we get
\[ \Rightarrow y = - 4\]
Thus, if \[x = 0\], then .
Therefore, the point of intersection of the line \[2x - y = 4\] and the equation \[y = 0\] is \[\left( {0, - 4} \right)\].
Hence, the equation \[2x - y = 4\] meets the \[y\]–axis at \[\left( {0, - 4} \right)\].
Thus, option (d) is also a correct option.
Hence option (c) and (d) both are correct answers.
Note: We have rejected options (a) and option (b). This is because the given equation is a linear equation in two variables. The graph of a linear equation in two variables is always a straight line. Thus, it cannot intersect another linear equation at two different points. Since the equation \[2x - y = 4\] meets the \[x\]–axis at \[\left( {2,0} \right)\], it cannot meet the same axis at \[\left( { - 4,0} \right)\]. Similarly, it cannot meet the \[y\]–axis at \[\left( {0,2} \right)\].
We can verify this by substituting the point into the equation.
Substituting \[x = 0\] and \[y = 2\] in \[2x - y = 4\], we get
\[\begin{array}{l} \Rightarrow 2\left( 0 \right) - 2 = 4\\ \Rightarrow - 2 = 4\end{array}\]
This is not possible. Thus, the point \[\left( {0,2} \right)\] does not lie on the line \[2x - y = 4\].
Substituting \[x = - 4\] and \[y = 0\] in \[2x - y = 4\], we get
\[\begin{array}{l} \Rightarrow 2\left( { - 4} \right) - 0 = 4\\ \Rightarrow - 8 = 4\end{array}\]
This is not possible. Thus, the point \[\left( { - 4,0} \right)\] does not lie on the line \[2x - y = 4\].