Daily Practice Problems

📝 SolutionCheck ratios: $a_1/a_2 = 2/4 = 1/2$, $b_1/b_2 = 3/6 = 1/2$, $c_1/c_2 = 7/14 = 1/2$. Since all three ratios are equal ($a_1/a_2 = b_1/b_2 = c_1/c_2$), the lines are coincident — they represent the same line, so there are infinitely many solutions.

📝 SolutionFor infinitely many solutions, $a_1/a_2 = b_1/b_2 = c_1/c_2$. Here $3/6 = 1/2$ and we need $1/k = 1/2$, so $k = 2$. Check: $c_1/c_2 = 5/10 = 1/2$ too, confirming $k=2$ works.

📝 SolutionAdding both equations: $(x+y) + (x-y) = 10 + 4 \Rightarrow 2x = 14 \Rightarrow x = 7$. Substituting back: $7 + y = 10 \Rightarrow y = 3$. The elimination method is fastest here since $y$ cancels directly on addition.

📝 SolutionNo solution means the lines never meet, which happens when they are parallel and distinct: $a_1/a_2 = b_1/b_2 \neq c_1/c_2$. This is the 'inconsistent' case in the pair of linear equations chapter.

📝 SolutionLet the numbers be $x$ and $y$ with $x + y = 18$ and $x - y = 4$. Adding: $2x = 22 \Rightarrow x = 11$, then $y = 18 - 11 = 7$. Always set up two equations from the word problem before solving — don't guess-and-check under exam time pressure.