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Chapter 1: Problem 55

Find an equation of the line passing through the points. Sketch the line, $$(5,-1),(-5,5)$$

Short Answer

Expert verified
The equation of the line passing through the points (5,-1) and (-5,5) is \(y = -0.6x + 2\).

Step by step solution

01

Find the Slope

First, let's calculate the slope of the line. Using the formula \(m = (y_2 - y_1) / (x_2 - x_1) = (5 - (-1)) / (-5 - 5) = -0.6.\)
02

Find the y-intercept

Next, let's find the y-intercept. To do this, use the slope and one of the given points and substitute into the line equation \(y = mx + c\). So, \(-1 = -0.6 * 5 + c\), solve for \(c\) to get \(c = 2\).
03

Write Line Equation

Now, we can write down the equation of the line. Given the slope \(-0.6\) and y-intercept \(2\), our line equation is \(y = -0.6x + 2\).

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Key Concepts

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

Understanding Slope
The slope of a line is a measure of its steepness and direction. It is often represented by the letter \(m\) in the line equation. To calculate the slope between two points, you use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). This formula represents the ratio of change in the y-values to the change in the x-values. In simpler terms, it tells you how much the line goes up (or down) for every step it moves to the right.
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal.
  • If the slope is undefined, the line is vertical.
In our exercise, the slope is computed as \(-0.6\), meaning for each step to the right, the line descends by 0.6 units.
Finding the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It is usually represented by the letter \(c\) in the line equation \(y = mx + c\). To find the y-intercept given a slope and a point through which the line passes, you substitute the slope \(m\) and the coordinates \((x, y)\) of one point into the line equation and solve for \(c\).
For example, using our exercise, we substitute the slope \(-0.6\) and the point \((5, -1)\) into the formula:
\(-1 = -0.6 \times 5 + c\)
Solving for \(c\), we find \(c = 2\). This means the line crosses the y-axis at \(2\). The y-intercept allows us to easily plot the line starting from the y-axis.
Constructing the Line Equation
To write the equation of a line, we use the form \(y = mx + c\) where \(m\) is the slope, and \(c\) is the y-intercept. Once we have determined both \(m\) and \(c\), we can construct the equation that fully describes our line.
From the previous calculations, we know:
  • The slope \(m\) is \(-0.6\).
  • The y-intercept \(c\) is \(2\).
Thus, the equation of our line is \(y = -0.6x + 2\). This equation can be used to compute the y-value for any given x-value along this line. It's a powerful tool in coordinate geometry as it defines the exact graph of the line.
Introduction to Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use algebraic techniques to study geometrical figures. It involves graphing equations to understand shapes, sizes, and other properties of geometrical figures in the coordinate plane.
Points on this plane are usually defined by coordinate pairs \((x, y)\). Basics include:
  • Finding distances between points.
  • Determining midpoints of lines.
  • Creating graphs of equations.
The fundamental elements, like the slope and y-intercept, help analyze and understand the behavior of lines and other shapes. They are key to translating linear equations into visual graphs, giving us a clearer insight into spatial relationships between different geometrical entities.

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