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.