Understanding Consistent System of Linear Equations

A system of linear equations is considered consistent if there exists at least one set of values for the variables that satisfies all the equations simultaneously. In this problem, we have a system of three linear equations with two variables, x and y:

1. \(\displaystyle 2k^2x + 3y - 1 = 0 \implies 2k^2x + 3y = 1\)
2. \(\displaystyle 7x - 2y + 3 = 0 \implies 7x - 2y = -3\)
3. \(\displaystyle 6kx + y + 1 = 0 \implies 6kx + y = -1\)

For a system of three linear equations in two variables to be consistent, the determinant of the matrix formed by the coefficients of x, coefficients of y, and the constant terms must be equal to zero. This condition arises because if the system is consistent, the three lines represented by the equations must intersect at a single point, or be coincident, or two lines are parallel and the third intersects them at distinct points, but for three distinct lines to intersect at one point, the area of the triangle formed by them would be zero, which relates to the determinant condition.

Setting up the Determinant for Consistency

We can write the coefficients and constant terms in a matrix form:

The determinant of this 3x3 matrix must be zero for the system to be consistent.

\(\displaystyle \begin{vmatrix} 2k^2 & 3 & 1 \\ 7 & -2 & -3 \\ 6k & 1 & -1 \end{vmatrix} = 0\)

Calculating the Determinant

Let's calculate the determinant. We can expand along the first row:

\(\displaystyle 2k^2 \begin{vmatrix} -2 & -3 \\ 1 & -1 \end{vmatrix} - 3 \begin{vmatrix} 7 & -3 \\ 6k & -1 \end{vmatrix} + 1 \begin{vmatrix} 7 & -2 \\ 6k & 1 \end{vmatrix} = 0\)

\(\displaystyle 2k^2 ((-2)(-1) - (-3)(1)) - 3 ((7)(-1) - (-3)(6k)) + 1 ((7)(1) - (-2)(6k)) = 0\)

\(\displaystyle 2k^2 (2 + 3) - 3 (-7 + 18k) + 1 (7 + 12k) = 0\)

\(\displaystyle 2k^2 (5) - (-21 + 54k) + (7 + 12k) = 0\)

\(\displaystyle 10k^2 + 21 - 54k + 7 + 12k = 0\)

Combine like terms:

\(\displaystyle 10k^2 + (-54k + 12k) + (21 + 7) = 0\)

\(\displaystyle 10k^2 - 42k + 28 = 0\)

Solving the Quadratic Equation for k

We have a quadratic equation in terms of k: \(10k^2 - 42k + 28 = 0\). We can simplify this equation by dividing all terms by 2:

\(\displaystyle 5k^2 - 21k + 14 = 0\)

We can solve this quadratic equation using the quadratic formula, which for an equation of the form \(ax^2 + bx + c = 0\) is given by:

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

In our equation, \(5k^2 - 21k + 14 = 0\), we have \(a=5\), \(b=-21\), and \(c=14\). Substituting these values into the quadratic formula to find k:

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

\(\displaystyle k = \frac{21 \pm \sqrt{441 - 280}}{10}\)

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

So, the values of k for which the system of equations is consistent are \(\frac{21 + \sqrt{161}}{10}\) and \(\frac{21 - \sqrt{161}}{10}\).

Revision Table: System Consistency and Determinants

Additional Information: Solving Systems of Equations

Solving systems of linear equations is a fundamental concept in algebra. For a system of 'm' equations in 'n' variables:

• If m < n, the system is usually underdetermined and may have infinite solutions or no solution.
• If m = n, the system can have a unique solution, infinite solutions, or no solution. Methods like substitution, elimination, matrix inversion, or Cramer's rule can be used. For Cramer's rule, the determinant of the coefficient matrix must be non-zero for a unique solution.
• If m > n, the system is overdetermined. As seen in this problem (3 equations, 2 variables), an overdetermined system is generally inconsistent unless the equations are dependent. For consistency in an overdetermined system, extra conditions (like the determinant condition used here for 3 equations/2 variables) must often be met.

In the case of three lines in a 2D plane (representing 3 equations in 2 variables), consistency means they intersect at a single point, or all three are the same line, or two are parallel and the third passes through them at distinct points but the specific determinant condition ensures they intersect at a single point unless specific dependencies exist.