Question

If \( x - 4y - 14 = 0 \) and \( 5x - y - 13 = 0 \) will have:

• A. unique solution
• B. no solution
• C. infinite number of solutions
• D. None of these

Hint:

If the system of equations has a unique solution, the lines represented by the equations intersect at one point. Use substitution or elimination methods to solve.

Answer 1

The Correct Option is A

Step 1: Write the given system of equations
We are given the system of equations: \[ x - 4y - 14 = 0 \quad \text{(Equation 1)} \] \[ 5x - y - 13 = 0 \quad \text{(Equation 2)} \] Step 2: Rearrange the equations
Rearrange the equations to make them easier to work with: \[ x - 4y = 14 \quad \text{(Equation 1 rearranged)} \] \[ 5x - y = 13 \quad \text{(Equation 2 rearranged)} \] Step 3: Solve the system using the substitution or elimination method
We will solve this system using the elimination method.
Multiply Equation 1 by 5 to align the coefficients of \( x \) in both equations: \[ 5(x - 4y) = 5 \times 14 \] \[ 5x - 20y = 70 \quad \text{(Equation 3)} \] Now subtract Equation 2 from Equation 3: \[ (5x - 20y) - (5x - y) = 70 - 13 \] \[ 5x - 20y - 5x + y = 57 \] \[ -19y = 57 \] Solve for \( y \): \[ y = \frac{57}{-19} = -3 \] Step 4: Substitute \( y = -3 \) into one of the original equations
Substitute \( y = -3 \) into Equation 1: \[ x - 4(-3) = 14 \] \[ x + 12 = 14 \] \[ x = 14 - 12 = 2 \] Step 5: Conclusion
The solution is \( x = 2 \) and \( y = -3 \), which means the system has a unique solution. Thus, the correct answer is option \( (1) \).