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Chapter 11: Problem 65

In the following exercises, graph by plotting points. $$ x+y=-3 $$

Short Answer

Expert verified
Points: (-5, 2), (-3, 0), (0, -3), (1, -4). Draw a line through these points.

Step by step solution

01

Rewrite the Equation

Rewrite the equation in a form that makes it easy to plot points. The equation is already given in standard form: \(x + y = -3\).
02

Create a Table of Values

Choose values for x and solve for y. For instance: \[ \begin{array}{cc} x & y \ \hline -5 & 2 \ -3 & 0 \ 0 & -3 \ 1 & -4 \ \end{array} \]
03

Plot the Points

Take the ordered pairs from the table of values: (-5, 2), (-3, 0), (0, -3), and (1, -4). Plot these points on a coordinate graph.
04

Draw the Line

Connect the plotted points with a straight line, extending it in both directions. This is the graph of the equation \(x + y = -3\).

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Key Concepts

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

plotting points
Plotting points is the method of placing points on a coordinate graph to visually represent their location. Each point is represented by an ordered pair \(x, y\). The first value in the pair is the x-coordinate, which specifies the point's horizontal position. The second value is the y-coordinate, which specifies the vertical position.
To plot points effectively, follow these steps:
  • Start at the origin \(0, 0\), where the x and y axes intersect.
  • Move horizontally to the x-coordinate value.
  • From that position, move vertically to the y-coordinate value.
  • Mark the point where you land.
For example, to plot the point \(-3, 0\):
  1. Start at the origin.
  2. Move three units to the left along the x-axis (since -3 is negative).
  3. Since the y-coordinate is 0, you don't move up or down.
Now you have successfully plotted the point \-3, 0\ on the graph.
standard form equation
A linear equation in standard form is written as \(Ax + By = C\), where A, B, and C are constants. This form is particularly useful for graphing because it allows easy identification of intercepts.
In standard form, you can create a table of values to find specific points to plot:
  • Choose values for x (or y).
  • Solve the equation for the corresponding y (or x).
For the equation \(x + y = -3\):
  1. Choose different x values, such as -5, -3, 0, and 1.
  2. Substitute these x values into the equation and solve for y.
You'll get pairs of points: (-5, 2), (-3, 0), (0, -3), (1, -4).
These pairs represent the points you will plot on the graph.
coordinate graph
A coordinate graph, known as a Cartesian plane, consists of two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. The point where they intersect is called the origin \(0,0\). Each point on the graph is represented as an ordered pair \(x, y\).
To graph a linear equation using plotted points:
  • Draw the x and y axes, label them, and mark equal units along each axis.
  • Plot the points obtained from the table of values on the graph.
  • Connect the points with a linear straight line.
For the equation \(x + y = -3\), after plotting points \((-5, 2)\), \((-3, 0)\), \((0, -3)\), and \((1, -4)\), draw a straight line through them. This line represents all solutions to the equation.
Keep in mind:
  • Every linear equation graphs as a straight line.
  • Each point on the line satisfies the equation.
  • The line extends infinitely in both directions.

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Most popular questions from this chapter

In the following exercises, graph each pair of equations in the same rectangular coordinate system. $$ y=2 x \text { and } y=2 $$

In the following exercises, graph each equation. $$ y=\frac{1}{4} x-2 $$

In the following exercises, identify the most convenient method to graph each line. $$ y=-1 $$

In the following exercises, for each ordered pair, decide a. if the ordered pair is a solution to the equation. b. if the point is on the line. \(y=\frac{2}{3} x-1\) (a) \((0,-1)\) (b) \((3,1)\) (c) \((-3,-3)\) (d) \((6,4)\)

In the following exercises, graph the line given a point and the slope. $$ (-3,4) ; m=-\frac{1}{3} $$
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