Problem 21
Question
For the following problems, graph the equations. $$ 0 x+2 y=2 $$
Step-by-Step Solution
Verified Answer
Answer: The graph of the equation 0x + 2y = 2 is a horizontal line passing through the point (0, 1) on the y-axis.
1Step 1: Convert the given equation to slope-intercept form (y=mx+b)
First, let's rewrite the equation in the form y=mx+b where m represents the slope and b represents the y-intercept:
$$
2y = 2
$$
Divide by 2 on both sides:
$$
y = 1
$$
The equation is now in the form y=mx+b where m=0 (no x terms, means slope is 0) and b=1. This tells us the line is horizontal and goes through the y-axis at y=1.
2Step 2: Find the y-intercept
We already found b=1, so the y-intercept is at the point (0, 1).
3Step 3: Plot the y-intercept and draw the line
Plot the y-intercept (0,1) on the graph. Since the slope (m) of the equation is 0 and we have a horizontal line, we can simply draw a straight horizontal line starting from the y-intercept through all points where y=1. This is the graph of the given equation $$0x + 2y = 2$$.
Key Concepts
Slope-Intercept FormHorizontal LineY-InterceptGraph Plotting
Slope-Intercept Form
The slope-intercept form of a linear equation is a popular method for expressing straight lines. The form is given by the equation \( y = mx + b \), where:
For example, in the given equation \( 2y = 2 \), after dividing by 2, we obtain \( y = 1 \). In this equation, there is no \( x \), suggesting that the slope \( m = 0 \). This means the line is horizontal. Understanding the slope-intercept form makes it easier to graph and analyze the properties of linear equations.
- \( m \) stands for the slope of the line.
- \( b \) is the y-intercept, which is where the line crosses the y-axis.
For example, in the given equation \( 2y = 2 \), after dividing by 2, we obtain \( y = 1 \). In this equation, there is no \( x \), suggesting that the slope \( m = 0 \). This means the line is horizontal. Understanding the slope-intercept form makes it easier to graph and analyze the properties of linear equations.
Horizontal Line
A horizontal line is a type of line that moves straight across the graph from left to right, parallel to the x-axis. Its slope, represented as \( m \) in the slope-intercept form \( y = mx + b \), equals zero.
Why is the slope zero? If you imagine walking along a perfectly level sidewalk, you aren't going uphill or downhill. That's what happens with a horizontal line — no vertical movement.
In the example \( y = 1 \), since there are no \( x \) terms, it shows itself as a horizontal line crossing the y-axis where y equals 1. This characteristic makes horizontal lines easy to graph, as they simply follow a flat path on the graph at a constant y value.
Why is the slope zero? If you imagine walking along a perfectly level sidewalk, you aren't going uphill or downhill. That's what happens with a horizontal line — no vertical movement.
In the example \( y = 1 \), since there are no \( x \) terms, it shows itself as a horizontal line crossing the y-axis where y equals 1. This characteristic makes horizontal lines easy to graph, as they simply follow a flat path on the graph at a constant y value.
Y-Intercept
The y-intercept is where a line crosses the y-axis of a graph. This point is crucial because it provides a starting location to plot the line. It can be found in the slope-intercept equation \( y = mx + b \).
For example, in the equation \( y = 1 \), the y-intercept is at \( b = 1 \). This means that the line crosses the y-axis at the point (0, 1).
Identifying the y-intercept helps in setting up your graph correctly and quickly. It's the anchor from which the rest of the line is drawn, especially in horizontal lines like in our example.
- \( b \) is the y-intercept.
For example, in the equation \( y = 1 \), the y-intercept is at \( b = 1 \). This means that the line crosses the y-axis at the point (0, 1).
Identifying the y-intercept helps in setting up your graph correctly and quickly. It's the anchor from which the rest of the line is drawn, especially in horizontal lines like in our example.
Graph Plotting
Graph plotting involves visually representing the equation of a line on a graph. It starts with understanding the form of the equation and ends with drawing the line based on calculated points.
Here’s a simple way to plot the equation \( y = 1 \):
This straightforward process simplifies working with linear equations. Graph plotting helps make sense of abstract equations by providing a visual representation, aiding in a better understanding of their behavior.
Here’s a simple way to plot the equation \( y = 1 \):
- First, identify the y-intercept, which is (0, 1) in this case.
- Plot the y-intercept point on the graph.
- Since the slope \( m = 0 \), draw a straight horizontal line through this point, parallel to the x-axis.
This straightforward process simplifies working with linear equations. Graph plotting helps make sense of abstract equations by providing a visual representation, aiding in a better understanding of their behavior.