Two lines are parallel if their slopes are equal. The equation of a line in slope-intercept form, is:
[tex]y=mx+b[/tex]
Where m is the slope of the line.
Write each equation in slope-intercept form to find if the slopes of each line are the same or not, which will tell us if they are parallel or not.
a)
First, isolate y from the first equation to write the equation in slope-intercept form:
[tex]\begin{gathered} 2y=3x-5 \\ \Rightarrow y=\frac{3}{2}x-\frac{5}{2} \end{gathered}[/tex]
Notice that the slope of the first line is 3/2. Next, isolate y from the second equation:
[tex]\begin{gathered} 8-y=-3x \\ \Rightarrow-y=-3x-8 \\ \Rightarrow y=3x+8 \end{gathered}[/tex]
The slope of the second line is 3.
Therefore, this pair of lines is NOT parallel.
b)
Notice that both equations are already written in slope-intercept form:
[tex]\begin{gathered} y=3 \\ y=-5 \end{gathered}[/tex]
Since the variable x does not appear in the equation, this means that the coefficient of x is 0 in both cases. Then, the slope of these two lines is 0.
Therefore, this pair of lines IS parallel.
