Question

1. in which system are the lines parallel?

a. \\(\

$$\begin{cases}2x - y = 3 \\\\ x + 2y = 3\\end{cases}$$

\\) b. \\(\

$$\begin{cases}2x + 3y = 5 \\\\ 6x + 9y = 1\\end{cases}$$

\\)
c. \\(\

$$\begin{cases}x - y = 10 \\\\ x + y = 10\\end{cases}$$

\\) d. \\(\

$$\begin{cases}3x - y = 1 \\\\ 4x + y = 2\\end{cases}$$

\\)

Explanation:

Step1: Convert to slope-intercept form

For a line $Ax+By=C$, slope $m=-\frac{A}{B}$.

Step2: Calculate slopes for Option A

Line1: $2x-y=3$, $m_1=-\frac{2}{-1}=2$
Line2: $x+2y=3$, $m_2=-\frac{1}{2}$

Step3: Calculate slopes for Option B

Line1: $2x+3y=5$, $m_1=-\frac{2}{3}$
Line2: $6x+9y=1$, $m_2=-\frac{6}{9}=-\frac{2}{3}$

Step4: Calculate slopes for Option C

Line1: $x-y=10$, $m_1=-\frac{1}{-1}=1$
Line2: $x+y=10$, $m_2=-\frac{1}{1}=-1$

Step5: Calculate slopes for Option D

Line1: $3x-y=1$, $m_1=-\frac{3}{-1}=3$
Line2: $4x+y=2$, $m_2=-\frac{4}{1}=-4$

Answer:

B.

$$\begin{cases} 2x + 3y = 5 \\ 6x + 9y = 1 \end{cases}$$