For what values of k is the system of equations 2k²x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?

Concept:

Consider three linear eqaution in two variable:

a₁x + b₁y + c_{1 = 0}

a₂x + b₂y + c₂ = 0

a₃x + b₃y + c₃ = 0

Condition for the consistency of three simultaneous linear equations in 2 variables:

​​​​\( \left| {\begin{array}{*{20}{c}} a_1&b_1&c_1\\ a_2&b_2&c_2\\ a_3&b_3&c_3 \end{array}} \right|=0\)

Formula for Quadratic equation:

ax² + bx + c = 0

x = \(\rm \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

Calculation:

2k2x + 3y - 1 = 0      ....(1)

7x - 2y + 3 = 0      ....(2)

6kx + y + 1 = 0      ....(3)

For consistency of given simultaneous equation,

\( \left| {\begin{array}{*{20}{c}} 2k^2&3&-1\\ 7&-2&3\\ 6k&1&1 \end{array}} \right|=0\)

⇒ 2k²(-2- 3) - 3(7 - 18k) - 1(7 + 12k) = 0

⇒ -10k² - 21 + 54k - 7 - 12k = 0

⇒ -10k² + 42k - 28 =  0

⇒ 5k2 - 21k + 14 =  0

By using the formula,

\(x=\rm \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

\(\Rightarrow k=\rm \frac{-(-21) \pm \sqrt{(-21)^{2} - 4(5)(14)}}{2\times 5}\)

\(\therefore k=\rm \frac{21 \pm \sqrt{161}}{10}\)