Question

For what value of $k$ , do the equations $3x - y + 8 = 0$ and $6x - ky = - 16$ represent coincident lines?
A.Solution of ${3^k} - 9 = 0$
B.Solution of ${2^k} - 8 = 0$
C. $2$
D. $3$

Answer 1

Hint : As we can see that the above equations are linear equations in two variables. An equation for a straight line is called a linear equation. We know that the standard form of linear equations in two variables is $Ax + By = C$ . With a pair of linear equations in two variables, the condition of coincident lines is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ .

Complete step-by-step answer :
Here we have the equations $3x - y + 8 = 0$ and $6x - ky = - 16$ . We can write the second equation as $6x - ky + 16 = 0$ .
Here we have ${a_1} = 3,{a_2} = 6,{b_1} = - 1,{b_2} = - k,{c_1} = 8,{c_2} = 16$ .
So by putting these in the formula we can write $\dfrac{3}{6} = \dfrac{{ - 1}}{{ - k}} = \dfrac{8}{{16}}$ .
We can write $\dfrac{3}{6}$ as $\dfrac{1}{2}$ and also $\dfrac{8}{{16}}$ can be written as $\dfrac{1}{2}$ . So we can write the expression as $\dfrac{1}{2} = \dfrac{1}{k} = \dfrac{1}{2}$ .
Since we can see that two of the values are the same, we can equate any one to find the value of $k$ .
So we can write $\dfrac{1}{2} = \dfrac{1}{k}$ . By cross multiplication it gives us $k = 2$ .
Now in the first option we have Solution of ${3^k} - 9 = 0$ . We can write it as ${3^k} = 9$ .
WE know that $9 = {3^2}$ , so by putting this in the equation we can write ${3^k} = {3^2}$ .
So by comparing we have $k = 2$ .
Hence the option (a) and (c) are correct.
So, the correct answer is “Option A and C”.

Note : We should know that coincident lines means that the line that lies upon each other. We have two pair of equations i.e. ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ , by comparing the given equations in questions with these we have calculated the value. We know that the slope intercept form of a linear equation is $y = mx + c$ ,where $m$ is the slope of the line and $b$ in the equation is the y-intercept and $x$ and $y$ are the coordinates of x-axis and y-axis , respectively.