MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesFor what value of $k$ will the following pair of linear equations have infinitely many solutions? $$kx + 3y = k - 3$$ $$12x + ky = k$$
Step-by-Step Solution
The given system of linear equations is:
- $kx + 3y - (k - 3) = 0$
- $12x + ky - k = 0$ \nComparing with standard form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:
- $a_1 = k$, $b_1 = 3$, $c_1 = -(k - 3)$
- $a_2 = 12$, $b_2 = k$, $c_2 = -k$ \nFor infinitely many solutions, the condition is: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$ \nSubstituting the values: $$\frac{k}{12} = \frac{3}{k} = \frac{-(k - 3)}{-k}$$ \nFrom $\frac{k}{12} = \frac{3}{k}$: $$k^2 = 36 \implies k = \pm 6$$ \nFrom $\frac{3}{k} = \frac{k - 3}{k}$: $$3 = k - 3 \implies k = 6$$ \nThe common value satisfying both equations is $k = 6$. \nHence, the correct option is A) $k = 6$.
Detailed Options Breakdown
Option : $k = 6$ (Correct Answer)
Correct choice. Refer to the step-by-step verified solution guidelines above for details.
Option 1: $k = -6$
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Pair of Linear Equations in Two Variables.
Option 2: $k = 0$
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Pair of Linear Equations in Two Variables.
Option 3: $k = 12$
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Pair of Linear Equations in Two Variables.
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