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Chapter 4: Problem 24

Graph each equation by finding the intercepts and at least one other point. $$x+3 y=0$$

Short Answer

Expert verified
To graph the equation \(x + 3y = 0\), we find the intercepts and another point. 1. x-intercept: Set \(y = 0\), and we get \(x = 0\). So, the x-intercept is (0, 0). 2. y-intercept: Set \(x = 0\), and we get \(y = 0\). So, the y-intercept is also (0, 0). 3. Another point: Choose \(x = 3\), and we get \(y = -1\). The point is (3, -1). Now, plot the points (0, 0), (3, -1) on a Cartesian plane and draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, we will set \(y = 0\) in the given equation and solve for \(x\). \(x + 3(0) = 0\) From this, we can see that the x-intercept is at the point (0, 0).
02

Find the y-intercept

To find the y-intercept, we will set \(x = 0\) in the given equation and solve for \(y\). \(0 + 3y = 0\) From this, we can also see that the y-intercept is at the point (0, 0).
03

Find at least one other point

To find another point on the line, we can choose any non-zero value for either x or y and solve for the other variable. Let's choose \(x = 3\). \(3 + 3y = 0\) To solve for \(y\), subtract 3 from both sides of the equation: \(3y = -3\) Now, divide by 3 to find the value of \(y\): \(y = -1\) This means when \(x = 3, y = -1\), giving us the point (3, -1).
04

Graph the equation

Now that we have the x-intercept (0, 0), the y-intercept (0, 0), and a third point (3, -1), we can plot these points on a Cartesian plane and draw a line through them to graph the given equation \(x + 3y = 0\).

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Key Concepts

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

Intercepts
When graphing a linear equation, intercepts are the points at which the graph crosses the x-axis and y-axis.
  • The x-intercept is found by setting the value of y to 0 and solving for x. This is where the line crosses the x-axis.
  • The y-intercept is determined by setting x to 0 and solving for y, representing where the line crosses the y-axis.
For the equation \[x + 3y = 0\], the x-intercept is found by substituting y with 0, resulting in the x-intercept at the point (0, 0).
Similarly, setting x to 0 gives us the y-intercept, which is also at (0, 0).
These intercepts provide crucial reference points for drawing the line, indicating where it intersects with the axes.
Graphing Equations
Graphing equations involves plotting points derived from the equation on a graph and then connecting them to form a line or curve. In the case of linear equations, the graph will always be a straight line.
  • Start by identifying the intercepts, since these give you initial points on the line.
  • Next, choose any other value for x or y, solve the equation, and plot this additional point. This helps to verify the direction of the line.
For the equation \[x + 3y = 0\], after finding the intercepts at (0, 0), we also find another point such as (3, -1), where x equals 3 and y equals -1.
Placing these points on the graph allows us to draw the line connecting all points, providing a visual representation of the equation.
Cartesian Plane
A Cartesian plane is a two-dimensional plane formed by the intersection of two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical line is the y-axis.
  • Each point on the plane is identified by a pair of numbers (x, y), known as coordinates.
  • The point where the axes intersect is called the origin, marked by the coordinate (0, 0).
For graphing equations like \[x + 3y = 0\], we use the Cartesian plane to plot all relevant points, such as intercepts and additional points derived from the equation.
Connecting these points illustrates the line that represents the equation on the plane.
Understanding the Cartesian plane is fundamental to grasping how graphs of linear equations are projected and interpreted.

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Most popular questions from this chapter

The velocity of an object, \(v\) (in feet per second), of an object during free- fall \(t\) sec after being dropped can be defined by the function $$ v(t)=32 t $$ a) Find the velocity of an object 1 sec after being dropped. b) Find the velocity of an object 3 sec after being dropped. c) When will the object be traveling at 256 ft / sec? d) Graph the function.

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