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(B) For a system of two linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ to have infinitely many solutions,the coefficients must satisfy t

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$6$

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(B) For a system of two linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ to have infinitely many solutions,the coefficients must satisfy the condition: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.Given equations are $2x + 3y = 5$ and $4x + ky = 10$.Here,$a_1 = 2, b_1 = 3, c_1 = 5$ and $a_2 = 4, b_2 = k, c_2 = 10$.Substituting these values into the condition: $\frac{2}{4} = \frac{3}{k} = \frac{5}{10}$.Simplifying the ratios: $\frac{1}{2} = \frac{3}{k} = \frac{1}{2}$.From $\frac{1}{2} = \frac{3}{k}$,we get $k = 3 	imes 2 = 6$.

Part $I$	Part $II$
$1.$ The solution of $2x - y = 4$ and $3x - 2y = 5$	$a.$ $x = 3, y = 2$
$2.$ The solution of $5x - y = 14$ and $4x - 3y = 9$	$b.$ $x = 3, y = 1$
$3.$ The solution of $4x - 3y = 5$ and $3x - y = 5$	$c.$ $x = 2, y = 1$
$4.$ The solution of $3x - 2y = -1$ and $2x - 5y = -8$	$d.$ $x = 1, y = 2$

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