Question

1. using the digits 1 – 9 at most ont time each, fill in the blanks so that two lines are parallel.

____ ( x + ) __ ( y = ) ____
____ ( x + ) __ ( y = ) ____

Explanation:

Step1: Recall parallel line condition

Parallel lines have equal slopes. Rewrite $Ax+By=C$ as $y=-\frac{A}{B}x+\frac{C}{B}$, so $\frac{A_1}{B_1}=\frac{A_2}{B_2}$ (simplified ratio).

Step2: Pick equal slope ratios

Choose $\frac{1}{2}$ and $\frac{3}{6}$ (simplify to $\frac{1}{2}$; digits 1,2,3,6 used).

Step3: Assign constants (unused digits)

Use remaining digits 4,5,7,8,9. Set constants: $1x+2y=7$, $3x+6y=21$ invalid (21 uses 1, repeat). Adjust to $1x+2y=5$, $3x+6y=15$ invalid. Try $\frac{2}{4}=\frac{1}{2}$: $2x+4y=9$, $1x+2y=5$. Check slopes: $-\frac{2}{4}=-\frac{1}{2}$, $-\frac{1}{2}=-\frac{1}{2}$ (equal). Digits used: 2,4,9,1,2 → repeat 2. Correct: $2x+4y=9$, $1x+2y=5$ invalid. Use $\frac{1}{3}=\frac{2}{6}$: $1x+3y=4$, $2x+6y=8$. Check slopes: $-\frac{1}{3}=-\frac{2}{6}=-\frac{1}{3}$. Digits: 1,3,4,2,6,8 (all unique, 1-9 no repeats).

Answer:

$1x + 3y = 4$
$2x + 6y = 8$

(Note: Valid alternative pairs exist, e.g., $3x+6y=9$, $1x+2y=4$, as long as slope ratios are equal and digits 1-9 are used at most once.)