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Chapter 2: Problem 11

Graph \(2 x+4=0\)

Short Answer

Expert verified
The graph for the linear equation \(2x + 4 = 0\) is a vertical line passing through the point (-2, 0).

Step by step solution

01

Solve for 'x'

First, isolate the variable 'x' by subtracting 4 from both sides of the equation \(2x + 4 = 0\), which gives \(2x = -4\). Now divide both sides by 2 to solve for 'x'. Therefore, 'x' equals to -2.
02

Plot the point (x= -2)

Plot the point on the number line where 'x' equals -2. This plot represents the solution of the equation \(2x + 4 = 0\). As this is a linear equation, the plot in a typical \(x-y\) graph would be a vertical line passing through 'x' equals -2.
03

Draw the line

Now that we have the necessary information, draw a vertical line passing through the point (-2, 0). This line is the graph that represents the equation \(2x + 4 = 0\).

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Key Concepts

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

Solving Linear Equations
Linear equations are mathematical expressions representing a straight line in a coordinate system. They usually have one variable, often denoted as 'x', and can be written in the standard form as \(ax + b = 0\). Solving a linear equation involves finding the value of the variable that makes the equation true.
In our example, we have the equation \(2x + 4 = 0\). Here are the steps to solve it:
  • First, we aim to isolate the variable 'x'. To do this, we subtract 4 from both sides: \(2x + 4 - 4 = 0 - 4\), resulting in \(2x = -4\).
  • Next, we divide both sides of the equation by 2 to solve for 'x': \(x = \frac{-4}{2} = -2\).
Thus, the solution for 'x' is -2. This value signifies the point on the x-axis where the line will intersect when graphed.
Coordinate System
The coordinate system is a two-dimensional plane used for graphing equations, consisting of horizontal (x-axis) and vertical (y-axis) lines intersecting at the origin point (0,0). Each point in this system is described by an ordered pair \((x, y)\).
When graphing linear equations such as \(2x + 4 = 0\), understanding how to use the coordinate system is essential.
  • The x-axis represents horizontal movement, assigning values to the variable 'x'.
  • The y-axis represents vertical movement, assigning values to 'y'.
  • The intersection at (0,0) is known as the origin, where both axes meet and begins the measurement of all points on the plane.
For our exercise, once we solved \(x = -2\), this determined the specific location on the x-axis where our vertical line would be positioned.
Graph Interpretation
Interpreting a graph involves understanding the visual representation of data, relationships, or equations on a coordinate system. In the context of our equation \(2x + 4 = 0\), the graph helps us see the solution visually.
Once we have isolated 'x' to solve it, plotting on the graph becomes easier. Here's how we interpret the graph for this equation:
  • The equation is linear, meaning the graph will depict a straight line.
  • Since we found \(x = -2\), the line is vertical as it does not depend on any 'y' value.
  • You plot the point at \((-2,0)\) on the graph, representing the solution.
  • Draw a line through this point that extends infinitely in both vertical directions.
This graph visually confirms that \(x = -2\) is the solution, as it shows exactly where the line intersects the x-axis. Such graphs are powerful tools for understanding equations and their respective solutions.

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