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Chapter 3: Problem 70

Find the equation of the line given two points on the line. (10,1) and (10,-3)

Short Answer

Expert verified
The equation of the line is \( x = 10 \).

Step by step solution

01

Recognize the Type of Line

Examine the given points (10,1) and (10,-3). Both points have the same x-coordinate, which indicates that the line passing through these points is vertical.
02

Determine the Equation of the Vertical Line

For vertical lines, the equation is simply the x-coordinate, since all points on the line have the same x. Therefore, the equation of the line passing through the points is \( x = 10 \).

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Key Concepts

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

Vertical Line
When dealing with lines on a graph, you might encounter vertical lines. These are special because they only go up and down, not side to side. All the points on a vertical line share the same x-coordinate. Let鈥檚 break down what this means.
  • A vertical line is a straight line that goes from top to bottom.
  • The x-coordinate is constant. This means that as you move up and down along the line, the x value doesn't change.
  • The equation of a vertical line is simple: it鈥檚 just in the form of x = a constant.
For example, for points (10,1) and (10,-3), both points have an x-coordinate of 10. Thus, the line passing through them is vertical and has the equation x = 10.
Points on a Line
When you have a line on a graph, you can identify it by using points. Let's go over the basics of how points contribute to the line's identity.
  • Each point on the line has an x and y coordinate, written as (x, y).
  • To describe a line, you need at least two points. They help to define the line鈥檚 location and direction.
  • In the case of vertical lines, these points only differ in their y-coordinates, because vertical lines maintain a constant x.
Using the example of points (10,1) and (10,-3), these points lie on the same vertical line, indicating that the x-value (10) stays the same, but their y-values differ, showing the vertical direction of the line.
x-coordinate
The x-coordinate in a graph indicates the horizontal position of a point. It's an essential part of understanding line equations.
  • The x-coordinate is the first number in an ordered pair (x, y), which shows how far left or right a point is on a graph.
  • In vertical lines, the x-coordinate remains constant, while different y-values occur.
  • This constancy is what lets us describe vertical lines with a straightforward equation: x = a number.
With our example, both points share an x-coordinate of 10: (10,1) and (10,-3). This is why the line is vertical, and its equation expresses this fixed x-value, giving us x = 10 as the equation.

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