Question

For \( a = \) ..............., the following simultaneous equations have an infinite number of solutions: \[ 10x + 13y = 6 \] \[ ax + 32.5y = 15 \]

Hint:

For infinite solutions in a system of linear equations, the coefficients of the variables and constants must be proportional.

Answer 1

Correct Answer: 25

Step 1: Condition for infinite solutions. 
For a system of linear equations to have an infinite number of solutions, the two equations must be dependent, i.e., they must be multiples of each other.

Step 2: Comparing the two equations. 
We compare the two equations: \[ 10x + 13y = 6 \] \[ ax + 32.5y = 15 \] For the equations to be multiples of each other, the ratios of the coefficients of \( x \), \( y \), and the constants must be equal. This gives the following system of equations: \[ \frac{10}{a} = \frac{13}{32.5} = \frac{6}{15} \]

Step 3: Solving for \( a \). 
First, simplify the ratio \( \frac{13}{32.5} \): \[ \frac{13}{32.5} = \frac{130}{325} = \frac{2}{5} \] Now, solve for \( a \) by equating \( \frac{10}{a} \) to \( \frac{2}{5} \): \[ \frac{10}{a} = \frac{2}{5} \] Cross-multiply: \[ 10 \times 5 = a \times 2 \Rightarrow a = 25 \]

Step 4: Conclusion. 
Thus, for the system to have infinite solutions, \( a = 25 \)