We can check if two lines are either parallel or perpendicular if we apply the following rule:
If two lines are parallel, then
[tex]\begin{gathered} \text{The slope of the two lines must be equal} \\ m_1=m_2 \end{gathered}[/tex]If two lines are perpendicular, then
[tex]\begin{gathered} \text{The two slopes will be related by} \\ m_1m_2=-1 \end{gathered}[/tex]So to check if the two lines are parallel
Step 1: Write the equation of a line in slope-intercept form
y=mx+c
Step2: Find the slopes of the equation
For the first line
[tex]\begin{gathered} 3x+7y=15 \\ 7y=-3x+15 \\ y=-\frac{3}{7}x+\frac{15}{7} \end{gathered}[/tex]So, the slope of the first line when compared to the general equation
[tex]m_1=-\frac{3}{7}[/tex]For the second line
[tex]\begin{gathered} 7x-3y=6 \\ 3y=7x-6 \\ y=\frac{7}{3}x-\frac{6}{3} \\ y=\frac{7}{3}x-2 \end{gathered}[/tex]So, the slope of the second line when compared to the general equation
[tex]m_2=\frac{7}{3}[/tex]The next step is to use the rule to confirm
Since
[tex]\begin{gathered} m_1=-\frac{3}{7} \\ m_2=\frac{7}{3} \\ \\ m_{1\text{ }}\times m_{2\text{ }}=-\frac{3}{7}\times\frac{7}{3}=-1 \end{gathered}[/tex]Since the product of their slopes = -1, then they are perpendicular
