The pair of linear equations $ 2x + 3y = 8 $ and $ 4x + 6y = 16 $ have:

Question

Examoogle · Accepted Answer

The correct answer is Option C: Infinitely many solutions. Check if the second equation is an exact multiple of the first by comparing the ratios of coefficients.

Step-by-Step Solution & Concepts:
Shortcut Trick: Check if the second equation is an exact multiple of the first by comparing the ratios of coefficients.

Formula Used: $$\Delta = \begin{vmatrix}2 & 3\\ 4 & 6\end{vmatrix}=0$$

Given: Equations: 2x + 3y = 8 and 4x + 6y = 16

Steps: Identify proportionality: Observe that the coefficients of the second equation are exactly twice those of the first: $4 = 2	imes2$, $6 = 2	imes3$, $16 = 2	imes8$.. Compute determinant: For a system $ax+by=c$ and $dx+ey=f$, the determinant $\Delta = ae - bd$ tells if the lines are parallel ($\Delta=0$) or intersecting ($\Delta 
eq 0$). Here $a=2$, $b=3$, $d=4$, $e=6$.. Conclude nature of solutions: Since $\Delta = 0$ and the constant terms are also in the same ratio (16 = 2×8), the equations represent the same line, giving infinitely many solutions.

Answer: C is correct because the two equations are proportional, representing the same line and thus have infinitely many solutions.

Exam Tip: In matrix form, a zero determinant indicates rank deficiency; similar checks are used for solving linear equations in higher dimensions.

Examoogle · Answer

Option A: Exactly one solution

Examoogle · Answer

Option B: No solution

Examoogle · Answer

Option D: Two distinct solutions

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)