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Chapter 1: Problem 26

Graph each equation by plotting points that satisfy the equation. $$2 x+y=-1$$

Short Answer

Expert verified
The graph of the equation \(2x + y = -1\) is a line passing through the points (-2,-3), (0,-1), and (2,-5).

Step by step solution

01

Generate values for x

This is a linear equation, therefore any real number can be substituted for x. Choose several numbers for x such as -2, 0, and 2 for variety.
02

Calculate corresponding y-values

Substitute each x-value into the equation and solve for y. When x = -2, y = \(2(-2) + y = -1\) which solves to y =-3. For x = 0, y = \(2(0) + y = -1\) which solves to y =-1. And when x = 2, y = \(2(2) + y = -1\) which solves to y =-5.
03

Plot the Points

Plot the points (-2,-3), (0,-1), and (2,-5) on a graph. These points should line up, as this is a linear equation.
04

Draw the Line

Sketch a line passing through all three points. This line is the graph of the given equation \(2x + y = -1\)

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Key Concepts

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

Graphing
Graphing refers to the visual representation of equations or data on a plane. It helps us see relationships between variables by displaying them as lines, curves, or shapes. For a linear equation, which is an equation of the form \(Ax + By = C\), graphing involves plotting points and then drawing a straight line passing through these points. This makes it easier to understand and analyze the behavior of the equation.

When graphing, the key steps include:
  • Identifying an equation.
  • Generating values for one variable.
  • Calculating corresponding values for the other variable.
  • Plotting these points on a coordinate plane.
  • Connecting the dots with a line.
By following these steps, you can create a clear graphical representation of the equation, which helps to visualize the solution set and analyze real-world scenarios.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by the intersection of two perpendicular lines called axes. The horizontal line is known as the x-axis, and the vertical line is known as the y-axis. These axes allow us to locate points on the plane using ordered pairs (x, y).

The point of intersection of the axes is called the origin, marked as (0, 0). From the origin, the axes extend infinitely in all directions, forming four quadrants:
  • Quadrant I: where both x and y are positive.
  • Quadrant II: where x is negative and y is positive.
  • Quadrant III: where both x and y are negative.
  • Quadrant IV: where x is positive and y is negative.
Understanding the structure of the coordinate plane is crucial as it provides the framework for plotting points and graphing equations like the one in the original exercise.
Plotting Points
Plotting points is a fundamental skill in graphing that involves placing specific points on a coordinate plane based on their coordinates. To plot a point, you use a pair of numbers known as an ordered pair, usually written in the form (x, y).

Here's how to plot points effectively:
  • Start at the origin (0, 0).
  • Move horizontally along the x-axis according to the x-coordinate.
  • Move vertically from your current position along the y-axis according to the y-coordinate.
  • Mark the point where you land; this is the location of your point on the plane.
When you plot all points for an equation, such as those found in our linear equation example, it should form a line if they are correctly plotted. This line graphically represents the equation and helps confirm the solution you have derived.

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Most popular questions from this chapter

Find \(f(-3)\) for \(f(x)=2 x^{2}-5 x-7 .[1.3]\).

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