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Chapter 3: Problem 32

Graph each linear equation. Plot four points for each line. $$x=-3$$

Short Answer

Expert verified
Plot points (-3, -2), (-3, 0), (-3, 2), and (-3, 4), then draw a vertical line through them.

Step by step solution

01

Understanding the Equation

The equation given is a vertical line where the value of x is always -3, regardless of the value of y.
02

Selecting Points for the Line

Choose any four values for y. We'll use y = -2, y = 0, y = 2, and y = 4 for simplicity.
03

Plot the Points

Plot the points (-3, -2), (-3, 0), (-3, 2), and (-3, 4) on a coordinate plane.
04

Draw the Line

Connect the points with a straight line. This line represents all points where x equals -3.

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Key Concepts

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

Vertical Lines
Vertical lines in a coordinate plane are unique because they have an undefined slope.
The equation of any vertical line will have the form of x = 'a constant value'. In our original exercise, the equation is $$x = -3$$.
This means no matter what the value of y is, x will always be -3.
Unlike other equations where both x and y values change, here only y changes while x remains fixed.
When drawing this line, you will notice it runs up and down through the x-coordinate -3.
So, vertical lines are straight lines that are parallel to the y-axis.
Coordinate Plane
The coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis).
These intersect at a point called the origin, denoted as (0,0). Each point on the plane is defined by an ordered pair (x, y). In the coordinate system, the x-value (or x-coordinate) indicates how far a point is to the right or left of the y-axis.
The y-value (or y-coordinate) shows how high or low the point is relative to the x-axis.
Together, these values help us locate points precisely in a 2D space.
Plotting Points
Plotting points involves placing them on the coordinate plane according to their (x, y) values.
For example, the point (-3, -2) is located 3 units to the left of the y-axis and 2 units down.
Likewise, (-3, 0) is 3 units left of the y-axis and right on the x-axis.
To plot points successfully:
  • Start at the origin (0,0)
  • Move left or right along the x-axis to the x-coordinate
  • Then move up or down to the y-coordinate
  • Mark the point on the grid
For our equation $$x = -3$$, since x is constant, all points will line up vertically along x = -3.
Algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols.
In the context of graphing linear equations, algebra helps us understand the relationship between x and y coordinates. For the equation$$x = -3$$, algebra tells us that:
  • 'x' is the independent variable
  • 'y' can be any real number
This means in our case, for any value of y, our x remains fixed at -3.
Understanding this concept leads to a better grasp of how we plot and interpret linear equations in terms of their graph on the coordinate plane.

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Most popular questions from this chapter

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has slope 5 and goes through \(\left(0, \frac{1}{2}\right)\).

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