CUET UG 2025 — Mathematics Algebra

For a system of equations to have infinitely many solutions, both equations must represent the same line. This occurs when the ratios of corresponding coefficients are equal:

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

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Given equations:

$$x + 2y = 5$$x+2y=5

$$3x + ky = 15$$3x+ky=15

Identifying coefficients:

For equation 1: $a_1 = 1$a1​=1, $b_1 = 2$b1​=2, $c_1 = 5$c1​=5

For equation 2: $a_2 = 3$a2​=3, $b_2 = k$b2​=k, $c_2 = 15$c2​=15

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Setting up the ratio equation:

$$\frac{1}{3} = \frac{2}{k} = \frac{5}{15}$$31​=k2​=155​

Simplifying the constant ratio:

$$\frac{5}{15} = \frac{1}{3}$$155​=31​

The ratios $\frac{1}{3} = \frac{1}{3}$31​=31​ match.

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Finding $k$k using the coefficient ratio:

$$\frac{1}{3} = \frac{2}{k}$$31​=k2​

$$k = 2 \times 3$$k=2×3

$$k = 6$$k=6

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Therefore, $k = 6$k=6.