Find the value of the constant k such that the system of equations has no solution:
$3x - 4y = 12$ and $kx + 8y = 20$
A system of two linear equations in two variables, such as \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\), has no solution if and only if the lines representing these equations are parallel but not coincident. Lines are parallel if the ratios of their coefficients for \(x\) and \(y\) are equal, i.e., \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\). For the lines to be distinct (not coincident), this ratio must not be equal to the ratio of their constant terms: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} eq \frac{c_1}{c_2}\). We are given the equations \(3x - 4y = 12\) and \(kx + 8y = 20\). Here, \(a_1 = 3\), \(b_1 = -4\), \(c_1 = 12\), and \(a_2 = k\), \(b_2 = 8\), \(c_2 = 20\).
To ensure the lines are parallel, we set the ratios of the coefficients of \(x\) and \(y\) to be equal:
\[\frac{3}{k} = \frac{-4}{8}\]
We simplify the right side of the equation:
\[\frac{3}{k} = -\frac{1}{2}\]
Multiplying both sides by \(2k\), we get:
\[6 = -k\]
\[k = -6\]
Now, we must verify that this value does not make the lines coincident by checking the constant ratio:
\[\frac{a_1}{a_2} = \frac{b_1}{b_2} = -\frac{1}{2}\]
\[\frac{c_1}{c_2} = \frac{12}{20} = \frac{3}{5}\]
Since \(-\frac{1}{2}
eq \frac{3}{5}\), the lines are indeed parallel and not coincident, which confirms that there is no solution for \(k = -6\).
Based on our analysis, the value of the constant \(k\) that results in a system with no solution is \(-6\). Therefore, the correct option is a.
\[\boxed{-6}\]
Imagine you are designing a map with two main bus routes. Route A is represented by the equation \(3x - 4y = 12\) and Route B is represented by \(kx + 8y = 20\). If you want to ensure these two bus routes never intersect—so that buses on Route A never cross paths with buses on Route B—you would need them to be parallel. By adjusting the factor \(k\), you can ensure that these lines stay perfectly parallel and never cross, preventing potential traffic congestion at a single point. This is a practical application of finding the slope relationships between linear equations.