CUET UG 2025 — Mathematics Algebra

For the system of linear equations $kx + 5y = 5$kx+5y=5 and $2x + 3y = 5$2x+3y=5 to be consistent, it must have at least one solution. A system is inconsistent when the lines are parallel but distinct (no solution).

For two equations $a_1x + b_1y = c_1$a1​x+b1​y=c1​ and $a_2x + b_2y = c_2$a2​x+b2​y=c2​, the system is inconsistent when:

$$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$a2​a1​​=b2​b1​​=c2​c1​​

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From the given equations:

$a_1 = k$a1​=k, $b_1 = 5$b1​=5, $c_1 = 5$c1​=5

$a_2 = 2$a2​=2, $b_2 = 3$b2​=3, $c_2 = 5$c2​=5

The ratios are:

$$\frac{a_1}{a_2} = \frac{k}{2}$$a2​a1​​=2k​

$$\frac{b_1}{b_2} = \frac{5}{3}$$b2​b1​​=35​

$$\frac{c_1}{c_2} = \frac{5}{5} = 1$$c2​c1​​=55​=1

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The system is inconsistent when:

$\frac{k}{2} = \frac{5}{3}$2k​=35​ and $\frac{k}{2} \neq 1$2k​=1

$$\frac{k}{2} = \frac{5}{3}$$2k​=35​

$$k = \frac{10}{3}$$k=310​

Since $\frac{10/3}{2} = \frac{5}{3} \neq 1$210/3​=35​=1, the condition is satisfied.

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The system is consistent for all values of $k$k except when the lines are parallel and distinct.

Therefore, the system is consistent when $k \neq \frac{10}{3}$k=310​