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Chapter 3: Problem 49

Find an equation of each line. See Example 7 . Parallel to \(x=0,\) through \((6,-8)\)

Short Answer

Expert verified
The equation of the line is \(x = 6\).

Step by step solution

01

Understand the problem

We are asked to find an equation of a line that is parallel to the line given by the equation \(x = 0\) and passes through the point \((6, -8)\).
02

Identify the characteristics of the line

The line \(x = 0\) is a vertical line that runs through the x-axis at \(x = 0\). A line parallel to it will also be vertical.
03

Equation of parallel line

A vertical line has an equation of the form \(x = a\) where \(a\) is a constant value representing the x-coordinate through which the line passes. In this case, the point through which it passes is \((6, -8)\), so \(a = 6\).
04

Write the equation of the line

Since the line is parallel to \(x = 0\) and passes through \((6, -8)\), its equation is \(x = 6\). This equation describes a line parallel to \(x = 0\) and vertical just like \(x = 0\), but passing through all points where the x-coordinate is 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
In geometry, understanding parallel lines is fundamental. Parallel lines are lines in a plane that do not intersect or cross each other at any point, regardless of how far they extend. You can picture them as train tracks extending infinitely.
  • They always maintain the same distance apart.
  • If one line is vertical, such that it follows a constant x-coordinate, any line parallel to it will also be a vertical line with a different fixed x-coordinate.
  • Parallelism ensures that vertical lines run in the same direction with no chance of intersection.
Knowing how lines interact in space helps in visualizing and solving geometry problems connected to lines and their equations.
Vertical Line
A vertical line is a type of line you encounter often in mathematics. Unlike horizontal lines, vertical lines rise straight up or down. They have unique properties:
  • Vertical lines have an infinite slope because they don鈥檛 move horizontally.
  • These lines are characterized by their unchanging x-coordinate at any given point along the line.
  • For instance, a vertical line at x = 0 would pass through all points on the plane where the x-coordinate is precisely 0.
Understanding a vertical line鈥檚 form is crucial for identifying its equation and its relation to points around it.
X-Coordinate
The x-coordinate is an essential element in understanding the position of a point on a 2-dimensional plane, forming one-half of what鈥檚 known as a coordinate pair (x, y). Here are a few important points to remember:
  • The x-coordinate represents a point's horizontal position on the plane.
  • If the x-coordinate is kept constant, the points will form a vertical line on the plane.
  • For example, in the point (6, -8), the x-coordinate is 6, indicating the point鈥檚 position six units away from the y-axis.
By focusing on the x-coordinate, you can determine the vertical position of a line, which is critical in deducing equations for vertical lines.
Vertical Lines Equation
The equation of a vertical line is straightforward but unique. Unlike other linear equations, which usually have the form y = mx + b, the equation of a vertical line does not involve the y-term:
  • The equation is expressed as x = a, where 'a' is the x-coordinate through which the line passes.
  • This simplification happens because the vertical line is the same for all y values at the specific x-coordinate.
  • In our exercise, since the line passes through the point (6, -8), its equation is x = 6, meaning all points on this line have x = 6 regardless of their y-value.
Being able to immediately recognize a vertical line's equation is an essential skill in solving and understanding problems related to the geometry of lines.

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Most popular questions from this chapter

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. Slope \(-2 ; y\) -intercept \((0,-4)\)

Answer each exercise with true or false. For the point \(\left(-\frac{1}{2}, 1.5\right),\) the first value, \(-\frac{1}{2},\) is the \(x\) -coordinate and the second value, \(1.5,\) is the \(y\) -coordinate.

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In \(2002,9.9\) million electronic bill statements were delivered and payment occurred. In 2005 , that number rose to 26.9 million. (Source: Forrester Research) CAN'T COPY THE IMAGE a. Write two ordered pairs of the form (years after \(2002,\) millions of electronic bills). b. Assume that this method of delivery and payment between the years 2002 and 2005 is linear. Use the ordered pairs from part (a) to write an equation of the line relating year and number of electronic bills. c. Use the linear equation from part (b) to predict the number of electronic bills to be delivered and paid in 2011 .

Solve each equation for \(y .\) See Section 2.5 \(10 x=-5 y\)

Find the slope of the line through the given points. \((-3.8,1.2)\) and \((-2.2,4.5)\)
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