Math Problem Statement

Use the given line and point to write an equation of a line that is parallel and perpendicular to the given equation.

Use the given line and point to write an equation of a line that is parallel and perpendicular to the given equation.

Solution

Let's solve the problem step by step. You are given the line equation y=3x6y = -3x - 6 and the point (7,2)(-7, 2). The task is to find:

  1. An equation of a line parallel to the given line that passes through the point (7,2)(-7, 2).
  2. An equation of a line perpendicular to the given line that passes through the point (7,2)(-7, 2).

Step 1: Find the slope of the given line

The given equation y=3x6y = -3x - 6 is in slope-intercept form y=mx+by = mx + b, where mm is the slope.
The slope of the given line is m=3m = -3.


Step 2: Equation of a line parallel to the given line

A line parallel to the given line will have the same slope, m=3m = -3.
Using the point-slope form of a line equation: yy1=m(xx1)y - y_1 = m(x - x_1) Substitute m=3m = -3, (x1,y1)=(7,2)(x_1, y_1) = (-7, 2): y2=3(x+7)y - 2 = -3(x + 7) Simplify: y2=3x21y - 2 = -3x - 21 y=3x19y = -3x - 19

The equation of the parallel line is:
y=3x19y = -3x - 19


Step 3: Equation of a line perpendicular to the given line

The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
The slope of the given line is m=3m = -3, so the slope of the perpendicular line is: m=13m = \frac{1}{3}

Using the point-slope form of a line equation: yy1=m(xx1)y - y_1 = m(x - x_1) Substitute m=13m = \frac{1}{3}, (x1,y1)=(7,2)(x_1, y_1) = (-7, 2): y2=13(x+7)y - 2 = \frac{1}{3}(x + 7) Simplify: y2=13x+73y - 2 = \frac{1}{3}x + \frac{7}{3} y=13x+73+2y = \frac{1}{3}x + \frac{7}{3} + 2 y=13x+133y = \frac{1}{3}x + \frac{13}{3}

The equation of the perpendicular line is:
y=13x+133y = \frac{1}{3}x + \frac{13}{3}


Final Answer:

  1. The equation of the parallel line is y=3x19y = -3x - 19.
  2. The equation of the perpendicular line is y=13x+133y = \frac{1}{3}x + \frac{13}{3}.

Would you like more details or additional explanation?


Related Questions:

  1. How can you verify that two lines are parallel or perpendicular?
  2. What is the general formula to find a line passing through a point with a given slope?
  3. How do we convert slope-intercept form to standard form?
  4. What happens if the given line is vertical or horizontal?
  5. How do we interpret the graph of these lines?

Tip:

To check if your equations are correct, substitute the given point into the equations to ensure they satisfy the line equations!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Point-Slope Form
Slope of a Line

Formulas

y - y1 = m(x - x1)
Slope of perpendicular line = -1 / (Slope of the original line)

Theorems

Parallel Line Theorem
Perpendicular Line Theorem

Suitable Grade Level

Grades 8-10