Question

For what value of k, the system of equations
$
  x + 2y = 5 \\
  3x + ky - 15 = 0 \\
$
Has a unique solution?

Answer 1

Hint – The given equations are of the form ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ , the system of equations will have a unique solution when $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ .
Complete step-by-step answer:
Given system of equations-
$
  x + 2y = 5 \\
  3x + ky - 15 = 0 \\
 $
The given equations are of the form ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ , where ${a_1} = 1,{b_1} = 2,{c_1} = - 5,{a_2} = 3,{b_2} = k,{c_2} = -15$
Now, the given system of equation will have unique solution if $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
So, this implies $
  \dfrac{1}{3} \ne \dfrac{2}{k} \\
   \Rightarrow k \ne 6 \\
 $
Thus, for all real values of k other than 6, the given equations will have a unique solution.
Note – Whenever such types of questions appear, then write down the equations given. Then write the values of the coefficients of the equation. Then, use the formula that the equations have unique solution if $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ , put the values of ${a_1} = 1,{b_1} = 2,{c_1} = - 5,{a_2} = 3,{b_2} = k,{c_2} = -15$ in the formula to find the value of k at which the system has a unique solution. As mentioned in the solution, after solving we get that for all real values of k other than 6, the given equations have a unique solution.