Problem 47 Graph the line containing the gi... [FREE SOLUTION]

Chapter 4: Problem 47

Graph the line containing the given point and with the given slope. $$(-2,-1) ; m=0$$

Short Answer

Expert verified

The given point is $(-2, -1)$ and the slope is $m = 0$. Using the point-slope form, we find the equation of the line to be $y = -1$. To graph the line, plot the y-intercept $(0, -1)$, and since the slope is 0, draw a horizontal line through the y-intercept.

Step by step solution

Use Point-Slope Form to Find the Line Equation

The point-slope form of a linear equation is given by: \[y - y_1 = m(x - x_1)\] where $(x_1, y_1)$ is the given point and $m$ is the slope. We are given $(-2, -1)$ as our point, which means $x_1 = -2$ and $y_1 = -1$. The slope, $m$, is given as $0$. Now plug these values into the point-slope formula: \[y - (-1) = 0(x - (-2))\]

Simplify the Equation

Now we can simplify the equation: \[y + 1 = 0(x + 2)\] Since 0 times any value is 0, the equation further simplifies to: \[y + 1 = 0\]

Solve for the Line Equation in Slope-Intercept Form

To find the final line equation, we need to solve for $y$ and put it in the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Subtract 1 from both sides of the equation: \[y = -1\] Now, we have our line equation: \[y = -1\]

Graph the Line

To graph the line, we can follow these steps: 1. Identify the slope, $m$, and the y-intercept, $b$, from the line equation. In this case, $m = 0$ and $b = -1$. 2. Plot the y-intercept, which is the point $(0, -1)$, on the graph. 3. Since the slope is 0, this means that the line is horizontal and all the points on the line will have the same y-coordinate. Therefore, every point on the line will have the form $(x, -1)$. 4. Draw a horizontal line through the y-intercept. Now, you have graphed the line containing the point $(-2, -1)$ and with the slope $m = 0$.

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Key Concepts

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

Point-Slope Form

Point-slope form is a way to represent the equation of a straight line, making use of a known point on the line and its slope. It's written as $ y - y_1 = m(x - x_1) $, where $(x_1, y_1)$ is a specific point on the line and $ m $ is the slope.
This form is very useful when you have specific coordinates and the slope of a line, allowing you to plug in these values directly.

To start, identify your point, in this case, $(-2, -1)$, translating to $x_1 = -2$ and $y_1 = -1$.
The slope $ m $ is given as $ 0 $ for this exercise.

By substituting these values into the point-slope form equation, we can derive the line equation. This form provides a convenient and precise way to establish the equation of linear graphs.

Slope-Intercept Form

The slope-intercept form of a linear equation is familiar and straightforward, written as $ y = mx + b $. This form elegantly showcases both the slope $ m $ and the y-intercept $ b $.
It's a go-to format for graphing since it clearly shows the starting point on the y-axis and how the line rises or falls.
Here鈥檚 how it ties into our example:

Once we had simplified the point-slope equation, we found it to be $ y = -1 $.
This directly fits the slope-intercept form, where the slope $ m = 0 $ and the y-intercept $ b = -1 $.

Since the slope is zero, we鈥檙e looking at a horizontal line, underlining the simplicity and directness of this equation form. Its strength is in clarity, allowing you to directly interpret the graph鈥檚 behavior.

Horizontal Line Equation

A horizontal line is one of the simplest forms, characterized by having a constant y-value across all points. The equation for a horizontal line in the coordinate plane takes the form $ y = b $, where $ b $ is a constant.
In our exercise, the equation $ y = -1 $ reflects a horizontal line:

The y-value remains consistent at $-1$ no matter what the x-value is.
Such a line indicates that the slope $ m $ is $ 0 $, meaning no vertical change as you move along the x-axis.

Graphing a horizontal line is straightforward:

You simply draw a straight line parallel to the x-axis at the specified y-coordinate.
All points $(x, -1)$ for any x reside on this line.

The beauty of horizontal lines lies in their simplicity, making it easy to visualize and understand.

Linear Graphs

Linear graphs depict equations of the first degree, typically in the form of straight lines across a chart. They are foundational in algebra and help in visualizing relationships between variables.
A simple way to understand linear graphs is through their basic properties:

They represent a constant rate of change, whether it's steep or flat, which is known as the slope.
Intercepts serve as key points, with the y-intercept showing where the graph crosses the y-axis.

In our specific example, the line equation $ y = -1 $ translates into a linear graph that doesn鈥檛 veer up or down, forming a flat, horizontal line across the y-value of $-1$.
The practice of plotting such graphs cements understanding of the relationship between algebraic expressions and their geometric representations on the coordinate plane.

91影视

Graph the line containing the given point and with the given slope. $$(-2,-1) ; m=0$$

Short Answer

Step by step solution

Use Point-Slope Form to Find the Line Equation

Simplify the Equation

Solve for the Line Equation in Slope-Intercept Form

Graph the Line

Key Concepts

Point-Slope Form

Slope-Intercept Form

Horizontal Line Equation

Linear Graphs

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