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Chapter 2: Problem 73

Exer. 73-74: Graph the lines on the same coordinate plane, and estimate the coordinates of the points of Intersection. Identify the polygon determined by the lines. $$2 x-y=-1 ; \quad x+2 y=-2 ; \quad 3 x+y=11$$

Short Answer

Expert verified
The polygon formed is a triangle with vertices at approximately \((-\frac{4}{5},\frac{2}{5})\), \((\frac{16}{5}, -\frac{13}{5})\), and \((2,5)\).

Step by step solution

01

Rearrange Equations into Slope-Intercept Form

First, we need to transform each of the given equations into the slope-intercept form, which is \( y = mx + b \).\1. For the first equation, \(2x - y = -1\): \[ y = 2x + 1 \]2. For the second equation, \(x + 2y = -2\): \[ y = -\frac{1}{2}x - 1 \]3. For the third equation, \(3x + y = 11\): \[ y = -3x + 11 \]
02

Graph the Lines

Using the slope-intercept form of each equation, plot the lines on a coordinate plane. Use the y-intercept and the slope to draw each line: 1. Line 1: Starts at (0,1) and rises 2 units for every 1 unit it runs to the right. 2. Line 2: Starts at (0,-1) and falls 1 unit for every 2 units it runs to the right. 3. Line 3: Starts at (0,11) and falls 3 units for every 1 unit it runs to the right.
03

Estimate Points of Intersection

Observe where the lines intersect each other on the graph:1. Intersection of Line 1 and Line 2: Solve \(2x + 1 = -\frac{1}{2}x - 1\), resulting in \(x = -\frac{4}{5}\), and \(y = \frac{2}{5}\).2. Intersection of Line 2 and Line 3: Solve \(-\frac{1}{2}x - 1 = -3x + 11\), resulting in \(x = \frac{16}{5}\), and \(y = -\frac{13}{5}\).3. Intersection of Line 1 and Line 3: Solve \(2x + 1 = -3x + 11\), resulting in \(x = 2\), and \(y = 5\).
04

Identify the Polygon

Now that you have found the intersection points, check how they connect:- The points of intersection are \((-\frac{4}{5},\frac{2}{5})\), \((\frac{16}{5}, -\frac{13}{5})\), and \((2,5)\).- Connect these points to form the polygon.- Since there are three sides formed by three intersecting lines, the resulting polygon is a triangle.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way of writing a linear equation, which allows you to easily see the slope and the y-intercept of the line. This form is presented as \( y = mx + b \), where:
  • \( m \) represents the slope of the line. It shows how steep the line is, indicating the rise (change in y) over the run (change in x).
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
Transforming a given equation into this form simplifies graphing, as it directly provides the slope and a starting point. In our exercise, equations are already converted to the form:
  • For \( 2x - y = -1 \), we have \( y = 2x + 1 \).
  • For \( x + 2y = -2 \), the form turns into \( y = -\frac{1}{2}x - 1 \).
  • Finally, \( 3x + y = 11 \) becomes \( y = -3x + 11 \).
Graphing Lines
With equations in slope-intercept form, graphing becomes straightforward. The y-intercept \( b \) tells us where to start on the y-axis. The slope \( m \) tells how many units to move up or down for each unit we move right.
  • For the equation \( y = 2x + 1 \), start at \( (0,1) \) and move 2 units up for 1 unit to the right.
  • \( y = -\frac{1}{2}x - 1 \) indicates starting at \( (0,-1) \) and moving 1 unit down for every 2 units to the right.
  • Lastly, \( y = -3x + 11 \) starts at \( (0,11) \) and moves down 3 units for every 1 unit to the right.
Drawing these lines on the coordinate plane provides a visual representation and helps in identifying intersection points.
Coordinate Plane
The coordinate plane is a two-dimensional space where we can plot points, lines, and shapes. It's divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
  • Each point is represented by a pair of values \((x, y)\).
  • The x-coordinate tells us how far the point is to the right or left of the y-axis.
  • The y-coordinate tells us how far up or down the point is from the x-axis.
For our graphing exercise, it's essential to correctly plot points and lines based on these coordinates. Each equation gives us precise starting points and directions to follow, making it easier to visualize where lines intersect.
Intersection Points
Intersection points are where two lines meet on a graph. They represent the solutions to a system of equations, showing where the conditions described by each equation are true simultaneously. Let's explore how we solve for these points:
  • Select two equations and set them equal to find their intersection.
  • For example, solving \( 2x + 1 = -\frac{1}{2}x - 1 \) gives \( x = -\frac{4}{5} \) and \( y = \frac{2}{5} \), marking one intersection point.
  • Continue with the other pairs: \(-\frac{1}{2}x - 1 = -3x + 11\) yields \( x = \frac{16}{5} \), \( y = -\frac{13}{5} \) and so on.
  • Intersections from \( 2x + 1 = -3x + 11 \) result in \( x = 2 \), \( y = 5 \).
These points are crucial for understanding the geometry of a plot since they reveal the vertices of the polygon formed by the intersecting lines. For this exercise, they form a triangle.

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