For what value of $k$ does the pair of linear equations $x + 2y - 3 = 0$ and $5x + ky + 7 = 0$ represent parallel lines?
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Question

For what value of $k$ does the pair of linear equations $x + 2y - 3 = 0$ and $5x + ky + 7 = 0$ represent parallel lines?

Accepted Answer

For parallel lines (no solution), the algebraic condition is:
$$\frac{a1}{a2} = \frac{b1}{b2} 
eq \frac{c1}{c2}$$

Comparing given equations with standard form:
- $a1 = 1, b1 = 2, c1 = -3$
- $a2 = 5, b2 = k, c2 = 7$

Now, set:
$$\frac{1}{5} = \frac{2}{k}$$

Cross-multiplying gives:
$$k = 10$$

Also, $\frac{2}{10} 
eq \frac{-3}{7}$, which holds true. Thus, $k = 10$.

Pair of Linear Equations in Two Variables