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Chapter 8: Problem 20

Write an equation of each line. See Examples 3 and \(4 .\) Slope \(0 ;\) through (-2,-4)

Short Answer

Expert verified
The equation of the line is \(y = -4.\)

Step by step solution

01

Understanding the Slope

The slope of the line is given as 0. A slope of 0 indicates a horizontal line. Hence, the equation of the line will be of the form \(y = c\) for some constant \(c.\)
02

Identify the Constant

Since the line passes through the point \((-2, -4)\) and is horizontal, every point on this line has the same \(y\)-value. Therefore, the constant \(c = -4.\)
03

Write the Equation

Now that we know the line is horizontal and passes through \(y = -4\), the equation of the line is \(y = -4.\)

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Key Concepts

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

Understanding Slope
In mathematics, the concept of the slope is crucial when discussing linear equations. The slope measures how steep a line is and the direction in which it tilts. To calculate the slope, we use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
where \( m \) stands for slope, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on a line.
A positive slope means the line rises as it moves from left to right, while a negative slope indicates a fall.
When the slope is zero, the line is perfectly horizontal, meaning there is no rise or fall regardless of changes in the x-value. In our exercise, the slope is 0, signaling a horizontal line.
Horizontal Line
A horizontal line is a straight line that goes from left to right or right to left, remaining parallel to the x-axis.
Every point on a horizontal line has the same y-coordinate. This characteristic means the slope of a horizontal line is always zero.
When writing the equation of a horizontal line, it takes the form:
  • \( y = c \)
where \( c \) is a constant that represents the y-coordinate of every point on the line.
In the given problem, since the line passes through the point \((-2, -4)\), every other point on the line also has a y-coordinate of \(-4\). Therefore, the equation is \( y = -4 \).
Graphing Linear Equations
Graphing linear equations helps visualize the relationship between variables. For linear equations, graphs are straight lines, characterized by constant slopes.
To graph a line, you first need two points. Once the slope is known, a single point is enough to draw the line since the slope tells you how to move from that point.
  • For a slope of 0, the line is horizontal, staying at the same y-level across all x-values.
  • The equation describes the line, like \( y = -4 \), which means every point \((x, y)\) on this line satisfies \( y = -4 \).
When graphing this in a coordinate plane, draw a straight line parallel to the x-axis at the height \(-4\). It shows all points where the y-value is \(-4\), irrespective of the x-value.

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Most popular questions from this chapter

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ \text { Find the domain of } f(x) \text { . } $$

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