Problem 20
Question
For the following problems, graph the equations. $$ 0 x+y=3 $$
Step-by-Step Solution
Verified Answer
Answer: The graph of the equation 0x + y = 3 represents a horizontal line passing through the y-axis at y = 3.
1Step 1: Identify the equation
The given equation is 0x + y = 3.
2Step 2: Rewrite the equation
Since 0x has no effect on the equation, we can rewrite it as y = 3.
3Step 3: Create a table of values
Let's create a table of values with x and y coordinates to help us plot the graph.
x | y
----|-----
-3 | 3
-2 | 3
-1 | 3
0 | 3
1 | 3
2 | 3
3 | 3
4Step 4: Plot the points on the graph
Now, using the table of values, plot the points on the graph.
5Step 5: Draw the line through the points
Once all points are plotted, draw a straight line passing through all of them. This line represents the graph of the equation 0x + y = 3, which is a horizontal line passing through the y-axis at y = 3.
Key Concepts
Equation of a LineCoordinate PlanePlotting Points
Equation of a Line
When you see a linear equation like the one given, understanding its structure is key. A linear equation, like 0x + y = 3, represents a straight line on the graph. In simpler terms, it's an equation where each solution corresponds to a point on this line.
The most common form of a linear equation is the slope-intercept form, written as \(y = mx + b\). Here, \(m\) is the slope, showing how steep the line is, and \(b\) indicates where the line crosses the y-axis. In our equation y = 3, you can see there's no \(x\) term, which means the slope \(m = 0\). This tells us that no matter the value of \(x\), \(y\) will always be 3.
With a horizontal line such as y = 3:
The most common form of a linear equation is the slope-intercept form, written as \(y = mx + b\). Here, \(m\) is the slope, showing how steep the line is, and \(b\) indicates where the line crosses the y-axis. In our equation y = 3, you can see there's no \(x\) term, which means the slope \(m = 0\). This tells us that no matter the value of \(x\), \(y\) will always be 3.
With a horizontal line such as y = 3:
- All points will have different x-coordinates but the same y-coordinate.
- The line will always be parallel to the x-axis.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is the backdrop for graphing equations like our example. It consists of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), intersecting at a point called the origin (0,0).
Each point on the coordinate plane is defined by an ordered pair (x, y), guiding you to its exact location. The x-value tells you how far to move right (positive) or left (negative) from the origin, while the y-value shows how far to move up (positive) or down (negative).
In practice:
Each point on the coordinate plane is defined by an ordered pair (x, y), guiding you to its exact location. The x-value tells you how far to move right (positive) or left (negative) from the origin, while the y-value shows how far to move up (positive) or down (negative).
In practice:
- A point like (2, 3) means move 2 units right and 3 units up from the origin.
- The coordinate plane is infinitely large, allowing for graphing all possible linear equations.
Plotting Points
Plotting points on a graph is a fundamental skill in graphing linear equations. It helps in constructing the visual representation of an equation.
Let's break down the process using our solution:
Once you've plotted these points on the coordinate plane, connect them with a line. For y = 3, you'll see that all the points lie on a horizontal line crossing the y-axis at 3, forming a perfect visual representation of the equation. This method can be used to graph any linear equation and is an effective way to understand points and lines!
Let's break down the process using our solution:
- Identify or calculate the y-value for different x-values, as shown in the table.
- Each pair of x and y forms a coordinate to plot on the graph.
- For our equation y = 3, you can select any x-value (like -3, -2, ..., 3) since they all yield y = 3.
Once you've plotted these points on the coordinate plane, connect them with a line. For y = 3, you'll see that all the points lie on a horizontal line crossing the y-axis at 3, forming a perfect visual representation of the equation. This method can be used to graph any linear equation and is an effective way to understand points and lines!