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Chapter 1: Problem 23

In Exercises 17鈥�30, write an equation for each line described. Passes through \((-12,-9)\) and has slope 0

Short Answer

Expert verified
The equation of the line is \( y = -9 \).

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of the equation of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Since the slope is 0, the line is horizontal.
02

Substitute Slope into Equation

Substitute the slope of 0 into the slope-intercept equation. This simplifies the equation to \( y = b \), because \( y = 0 \cdot x + b \).
03

Use Point to Find the Y-Intercept

Substitute the point \((-12, -9)\) into the equation to find \( b \). Since the line is horizontal, the value of \( y \) at any point on the line will be constant. Thus, \( b = -9 \). This gives us \( y = -9 \).
04

Write the Final Equation

The final equation of the line is simply \( y = -9 \). This is because a horizontal line means \( y \) is constant, irrespective of \( x \).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a standard way to express the equation of a straight line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, which tells you how steep the line is. The \( b \) is the y-intercept, which is where the line crosses the y-axis. Understanding this form helps you quickly identify these two important features of a line.
  • Slope \( (m) \): The change in \( y \) for a one-unit change in \( x \). If the slope is positive, the line rises as it moves to the right. If it is negative, the line falls. A slope of zero indicates a horizontal line, meaning there is no vertical change as \( x \) changes.
  • Y-Intercept \( (b) \): The value of \( y \) when \( x \) is zero. This point is always on the y-axis.
When you know the slope and y-intercept, you can easily sketch the line or understand its direction and position on a graph.
Horizontal Line
A horizontal line is a straight line where all points have the same y-coordinate. This means that the slope of a horizontal line is always zero, since there鈥檚 no vertical change as you move along the line.
  • Characteristics:
    • Flat and parallel to the x-axis.
    • Equation form: \( y = b \), where \( b \) is a constant.
    • Every point on the line shares the same y-value.
  • Example: For a horizontal line passing through \((-12, -9)\), the y-coordinate \(-9\) is constant for all points. Therefore, the equation is \( y = -9 \).
This simplicity makes calculations straightforward but also requires understanding that the x-coordinate can vary across all real numbers without affecting the y-coordinate.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept.
  • Importance: The y-intercept offers a starting point for graphing a line on a coordinate plane, crucial for understanding where the line will start along the y-axis.
  • Finding the Y-Intercept:
    • Substitute \( x = 0 \) into the line's equation to solve for \( y \). This y-value will be the y-intercept.
    • If the line's equation is in a simplified form like \( y = c \) (for horizontal lines), \( c \) itself is the y-intercept.
In our example, since the equation of the horizontal line is \( y = -9 \), the y-intercept is \(-9\). This means the line intersects the y-axis at \( (0, -9) \). Knowing the y-intercept lets you position the line correctly on the graph.

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