Solution:
a) Given the equation of line m below
[tex]y=\frac{2}{3}x-4[/tex]
Using a graphing tool, the graph of line m is shown below
The line is labeled m
b) A line, p, is parallel to line m and passes through point (3, 4)
The slope of parallel lines is equal.
Thus, the slope of line p is 2/3
To find the equation of line p, the formula is
[tex](y-y_1)=m(x-x_1)[/tex]
Substitute the coordinates and the slope into the formula above
[tex]\begin{gathered} (y-4)=\frac{2}{3}(x-3) \\ y-4=\frac{2}{3}x-2 \\ Collect\text{ like terms} \\ y=\frac{2}{3}x-2+4 \\ y=\frac{2}{3}x+2 \end{gathered}[/tex]
The graph of line p, is shown below
c) The equation of line p in slope-intercept form is
[tex]y=\frac{2}{3}+2[/tex]
d) Line s is perpendicular to line m,
To find the slope of a perpendicular line, the formula is
[tex]\begin{gathered} m_1\cdot m_2=-1 \\ Where \\ The\text{ slope of line }m,\text{ }m_1=\frac{2}{3} \\ The\text{ slope of line s is }m_2 \\ \frac{2}{3}\cdot m_2=-1 \\ m_2=-\frac{3}{2} \end{gathered}[/tex]
And line s passes through the point (0, -4),
The equation of line will be
[tex]\begin{gathered} (y-y_1)=m_2(x-x_1) \\ y-(-4)=-\frac{3}{2}(x-0) \\ y+4=-\frac{3}{2}x \\ Collect\text{ like terms} \\ y=-\frac{3}{2}x-4 \end{gathered}[/tex]
The graph of line s is shown below
e) The equation of line s in slope intercept form is
[tex]y=-\frac{3}{2}x-4[/tex]
f) To justify if line s is perpendicular to line p,
Using their slopes
[tex]\begin{gathered} Slope\text{ of line }s,\text{ }m_1=-\frac{3}{2} \\ Let\text{ }m_2\text{ be the slope of line p} \\ For\text{ perpendicular lines} \\ m_1\cdot m_2=-1 \\ -\frac{3}{2}\cdot m_2=-1 \\ \frac{-3m_2}{2}=-1 \\ -3m_2=-2 \\ m_2=\frac{-2}{-3}=\frac{2}{3} \\ The\text{ slope of }m_2=\frac{2}{3} \end{gathered}[/tex]
Hence, line s is perpendicular to line p
