Question

The pair of linear equations \(x+2y-5=0\) and \(3x+12y-10=0\) has

• A. no solution
• B. two solutions
• C. unique solution
• D. infinitely many solutions

Answer 1

The Correct Option is C

Given equations:

\(x + 2y - 5 = 0\)  ...(1)

\(3x + 12y - 10 = 0\)  ...(2) 

Step 1: Convert the equations into the standard form

Comparing with \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), we get:

\(a_1 = 1, b_1 = 2, c_1 = -5\)

\(a_2 = 3, b_2 = 12, c_2 = -10\)

Step 2: Check the consistency condition

We calculate the ratios:

\(\frac{a_1}{a_2} = \frac{1}{3}\),   \(\frac{b_1}{b_2} = \frac{2}{12} = \frac{1}{6}\),   \(\frac{c_1}{c_2} = \frac{-5}{-10} = \frac{1}{2}\)

Step 3: Analyze the results

Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the given pair of equations has a unique solution.

Final Answer: Unique solution

Answer 2

To determine the number of solutions for the system of linear equations \(x+2y-5=0\) and \(3x+12y-10=0\), we will analyze their slopes and intercepts. 
These are both linear equations in two variables, which can be compared using their coefficients.

The general form of a linear equation is \(Ax + By + C = 0\). Comparing both equations:

• First equation: \(x + 2y - 5 = 0\) 
  Comparing coefficients:\
  • \(A_1 = 1\)
  • \(B_1 = 2\)
  • \(C_1 = -5\)
• Second equation: \(3x + 12y - 10 = 0\)
  • \(A_2 = 3\)
  • \(B_2 = 12\)
  • \(C_2 = -10\)

To check if lines are parallel, coincident, or intersecting, use the conditions based on coefficients:

• If \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\), the lines are parallel with no solution.
• If \(\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}\), they coincide with infinitely many solutions.
• If \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\), they intersect at a unique point with a unique solution.

Calculate the ratios of the coefficients:

• \(\frac{A_1}{A_2} = \frac{1}{3}\)
• \(\frac{B_1}{B_2} = \frac{2}{12} = \frac{1}{6}\)
• \(\frac{C_1}{C_2} = \frac{-5}{-10} = \frac{1}{2}\)

Since \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\), the lines are not parallel and intersect at exactly one point, thus the system has a unique solution.