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Chapter 3: Problem 35

Graph each linear equation. Plot four points for each line. $$y+3=0$$

Short Answer

Expert verified
Graph a horizontal line where y = -3.

Step by step solution

01

Rewrite the Equation

Rewrite the equation in a more familiar form. Given equation is: y + 3 = 0
02

Solve for y

Isolate y by subtracting 3 from both sides of the equation: y = -3
03

Understand the Line

Recognize that the equation y = -3 represents a horizontal line where the y-coordinate is always -3 regardless of the x-coordinate.
04

Plot Points

Choose four values for x and find the corresponding y values. Since y = -3, it will remain the same for all points: (x, y) = (0, -3) (x, y) = (1, -3) (x, y) = (-1, -3) (x, y) = (2, -3)
05

Graph the Line

Plot the points (0, -3), (1, -3), (-1, -3), and (2, -3) on the graph and draw a straight line through these points.

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Key Concepts

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

linear equations
Linear equations are mathematical expressions showing a relationship between two variables, usually x and y, where the plot creates a straight line. The general form of a linear equation is presented as:
\[ax + by = c\]
The components of these equations are:
  • The coefficients a and b, which represent the slope and direction of the line.
  • The constant c, determining the line's position on the graph.
Solving linear equations involves finding values for x and y that satisfy the equation. This often means rearranging the equation to isolate one variable on one side. Understanding the nature of these equations is fundamental for graphing lines effectively.
plotting points
Plotting points on a coordinate plane is an essential skill in graphing linear equations. Each point is defined by an ordered pair \((x, y)\), where x is the horizontal position, and y is the vertical position. To plot a point:
  • Start at the origin \((0,0)\).
  • Move along the x-axis to the x-coordinate.
  • Then, move parallel to the y-axis to the y-coordinate.
For the provided equation, since
\[y = -3\],
it means no matter what x is, y will always be -3, creating points like \((0, -3)\), \((1, -3)\), \((-1, -3)\), and \((2, -3)\). Understanding the relationship between x and y values is the first step in visualizing how the line will look on the graph.
horizontal lines
Horizontal lines have a unique property: the y-value stays constant while the x-value can vary. The equation of a horizontal line can be written as
\[ y = k \]
where k is the constant y-value for all points along the line. In the exercise example, the equation
\[ y = -3 \]
tells us that the horizontal line passes through all points where y is -3. This means every point looks like
\((x, -3)\)
with x taking any value. Such horizontal lines are parallel to the x-axis and do not have an incline or slope, making them straightforward to plot and identify.
coordinate plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface used to graph equations. It consists of two perpendicular axes:
  • The horizontal axis (x-axis).
  • The vertical axis (y-axis).
The point where the axes intersect is called the origin, marked as \((0,0)\). Points are plotted based on their x and y coordinates, with each point represented as \((x, y)\). For example, to graph the equation
\[ y = -3 \],
we locate points where y is -3:
  • Start at the origin.
  • Move horizontally to each x-coordinate.
  • Then move vertically down to y = -3.
By connecting these points, you visualize the linear relationship defined by the equation. Understanding the layout of the coordinate plane is crucial for accurately plotting and graphing mathematical equations.

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Most popular questions from this chapter

Discussion Explain how to write an absolute value inequality as a compound inequality.

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