Understanding the System of Equations

We are given a system of two linear equations:

• Equation 1: \(2x - 3y - 5 = 0\)
• Equation 2: \(15y - 10x + 50 = 0\)

Rewriting Equations in Standard Form

To analyze the system, let's rewrite both equations in the standard form \(Ax + By = C\).

• Equation 1 can be written as: \(2x - 3y = 5\) Here, \(A_1 = 2\), \(B_1 = -3\), and \(C_1 = 5\).
• Equation 2 can be rearranged as: \(-10x + 15y = -50\) Here, \(A_2 = -10\), \(B_2 = 15\), and \(C_2 = -50\).

Determining the Nature of Solutions

We can determine if a system of linear equations \(A_1x + B_1y = C_1\) and \(A_2x + B_2y = C_2\) has a unique solution, infinitely many solutions, or no solution (inconsistent) by comparing the ratios of their coefficients:

• Unique Solution: If \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\). The lines intersect at a single point.
• Infinitely Many Solutions: If \(\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}\). The lines are coincident (the same line).
• Inconsistent (No Solution): If \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\). The lines are parallel and distinct.

Analyzing the Coefficient Ratios

Let's calculate the ratios for the given system:

• Ratio of x-coefficients: \(\frac{A_1}{A_2} = \frac{2}{-10} = -\frac{1}{5}\)
• Ratio of y-coefficients: \(\frac{B_1}{B_2} = \frac{-3}{15} = -\frac{1}{5}\)
• Ratio of constant terms: \(\frac{C_1}{C_2} = \frac{5}{-50} = -\frac{1}{10}\)

Now, let's compare these ratios:

We observe that \(\frac{A_1}{A_2} = -\frac{1}{5}\) and \(\frac{B_1}{B_2} = -\frac{1}{5}\). So, \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\).

However, when we compare this to the ratio of the constant terms, we see that \(-\frac{1}{5} \neq -\frac{1}{10}\). Therefore, \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\).

Conclusion

According to the conditions for analyzing systems of linear equations, the relationship \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\) indicates that the system of equations is inconsistent. This means there are no values of \(x\) and \(y\) that can satisfy both equations simultaneously, as the lines represented by the equations are parallel.