Question

The pair of linear equations \(5x - 4y + 8 = 0\) and \(7x + 6y - 9 = 0\) is

• A. consistent
• B. inconsistent
• C. dependent
• D. none of these

Hint:

The quickest way to check is to compare \(\frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\). If they are not equal, the system is consistent with a unique solution. You don't need to check the 'c' ratio.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A pair of linear equations can be classified based on its number of solutions:
- Consistent: The system has at least one solution (one or infinitely many).
- Inconsistent: The system has no solution.
- Dependent: The system has infinitely many solutions (a sub-category of consistent).
We can determine this by comparing the ratios of the coefficients \(a\), \(b\), and \(c\).

Step 2: Detailed Explanation:
The given equations are:
1. \(5x - 4y + 8 = 0\), where \(a_1 = 5, b_1 = -4, c_1 = 8\).
2. \(7x + 6y - 9 = 0\), where \(a_2 = 7, b_2 = 6, c_2 = -9\).
Let's compare the ratios of the coefficients of x and y:
\[ \frac{a_1}{a_2} = \frac{5}{7} \] \[ \frac{b_1}{b_2} = \frac{-4}{6} = -\frac{2}{3} \] Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) (\(\frac{5}{7} \neq -\frac{2}{3}\)), the lines representing these equations will intersect at a single, unique point.
A system with one solution is defined as being consistent.

Step 3: Final Answer:
The pair of linear equations is consistent.