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Chapter 12: Problem 45

Find the equation of the line through the given points. $$(1,5) \text { and }(-6,5)$$

Short Answer

Expert verified
The equation of the line through the points (1,5) and (-6,5) is y = 5.

Step by step solution

01

- Find the Slope (m)

To calculate the slope, use the formula \((y_2 - y_1) / (x_2 - x_1)\). The coordinates of the two points are (1,5) and (-6,5). Hence,\[ m = (5 - 5) / (1 - (-6)) = 0 \]As expected, the slope is 0.
02

- Find the Y-Intercept (b)

Because this is a horizontal line, it intercepts y-axis (where x is 0) at y-coordinate of the given points. Hence, b = 5.
03

- Write Down the Equation

Combine the slope (m) and y-intercept (b) to form the line equation y = mx + b. Upon substitution of m = 0 and b = 5, we get y = 5.

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Key Concepts

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

Slope of a Line
When we talk about the slope of a line, we're referring to how steep the line is. It's a measure of how much the y-coordinate changes for a change in the x-coordinate. You can think of it as rise over run, where "rise" is the change in the y-values and "run" is the change in the x-values.
To find the slope (\( m \)), use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • If the slope is positive, the line goes upwards as you move from left to right.
  • If the slope is negative, the line goes downwards from left to right.
  • If the slope is zero, the line is horizontal, meaning there is no rise at all.
For the points \((1,5)\) and \((-6,5)\), since \(y_2 - y_1 = 0\), the slope is 0, making it a horizontal line.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.
To find the y-intercept, substitute the known slope and one of the coordinates into the equation of a line: \(y = mx + b\).
In this scenario:
  • Because the line is horizontal, any point on the line will have the same y-coordinate.
  • For the line passing through \((1,5)\) and \((-6,5)\), the y-intercept \(b\) is 5.
This means the entire line is simply \(y = 5\), as the slope is zero and does not contribute to changing the y-value.
Horizontal Line
A horizontal line is a straight line that runs left to right and is parallel to the x-axis. This type of line has a special characteristic:
its slope is always zero. The reason for this is because there is no change in y as x changes.
Here are a few properties of horizontal lines:
  • All the points along a horizontal line share the same y-coordinate.
  • They can be represented by an equation in the form \(y = c\), where \(c\) is a constant.
The equation of the line through the points \((1,5)\) and \((-6,5)\) is \(y = 5\), reflecting the horizontal nature of the line.
Coordinates
Coordinates are a set of values that show an exact position on a graph. They come in an ordered pair, \((x, y)\).
The "x" part shows how far along the line you go horizontally from the origin, while the "y" part shows how far up or down you go.
  • The origin point on the graph is \((0,0)\).
  • In the exercise, we used the coordinates \((1,5)\) and \((-6,5)\) to calculate the slope and equation of the line.
Understanding how to plot and interpret coordinates helps in identifying the position and relationship between other points on the graph.

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Most popular questions from this chapter

Determine whether there is a line that contains all of the given points. If so, find the equation of the line. $$(-2,-4),(0,-3),(4,-1)$$

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