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Answer:

[tex]\Large\boxed{\text{Perpendicular Equation:} \ y = -\frac{2}{7}x + \frac{22}{7}}[/tex]

[tex]\Large\boxed{\text{Parallel Equation:} \ y = \frac{7}{2}x -12}[/tex]

Step-by-step explanation:

In order to find the equations that are parallel/perpendicular to the line [tex]y = \frac{7}{2}x - 4[/tex], we need to note a couple things about the relationships between lines and their parallel/perpendicular lines.

  • A) If a line is perpendicular to another, the slopes will be opposite reciprocals (for instance [tex]-2[/tex] and [tex]\frac{1}{2}[/tex] - multiplied, they equal -1.)
  • B) If a line is parallel to another, they will have the exact same slope.

Perpendicular:

We know that the slope of a perpendicular line will be the the opposite reciprocal of the line we're comparing it to.

Since the slope of our base line is [tex]\frac{7}{2}[/tex], we can find the reciprocal, then the opposite of that.

  • Reciprocal of [tex]\frac{7}{2} = \frac{2}{7}[/tex]
  • Opposite of [tex]\frac{2}{7} = -\frac{2}{7}[/tex]

So the slope of this line will be [tex]-\frac{2}{7}[/tex], making our equation [tex]y = -\frac{2}{7}x+ b[/tex]

However, y-intercepts will not stay the same. In order to find this, we can substitute the point (4, 2) into our equation to solve for b.

  • [tex]2 = -\frac{2}{7} \cdot 4 + b[/tex]
  • [tex]2 = -\frac{8}{7} + b[/tex]
  • [tex]b = 2+\frac{8}{7}[/tex]
  • [tex]b = 2 \frac{8}{7}[/tex]
  • [tex]b=\frac{22}{7}[/tex]

Now we know the y-intercept of this equation is [tex]\frac{22}{7}[/tex]. We can now finish off our equation of the line by substituting that in to what we already have,  [tex]y = -\frac{2}{7}x+ b[/tex].

[tex]y = -\frac{2}{7}x+ \frac{22}{7}[/tex]

Parallel:

As mentioned earlier, parallel lines will have the exact same slope but not the same y-intercept. Since the slope of our original equation is [tex]\frac{7}{2}[/tex], the slope for this one will also be [tex]\frac{7}{2}[/tex].

So we now know the equation looks something like [tex]y = \frac{7}{2}x + b[/tex].

In order to solve for b, we apply the same logic we did in the perpendicular line and substitute in the point (4, 2) into the equation.

  • [tex]2 = \frac{7}{2} \cdot 4 + b[/tex]
  • [tex]2 = \frac{28}{2}+b[/tex]
  • [tex]2 = 14+b[/tex]
  • [tex]b = 2-14[/tex]
  • [tex]b =-12[/tex]

Now that we know the slope and the y-intercept, we can finish off our equation as [tex]y = \frac{7}{2}x -12[/tex].

Hope this helped!

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