Problem 32 - Algebra for College Students

Question

Contains the point \((-3,-7)\) and is parallel to the \(x\) axis

Step-by-Step Solution

Answer

The equation is \(y = -7\).

Step 1: Understanding the Problem

To find the equation of a line that is parallel to the x-axis and passes through a given point We have the point \((x_1, y_1) = (-3, -7)\). A line parallel to the x-axis means it has no slope, i.e., it's horizontal.

Step 2: Determining the Equation

A horizontal line passing through a given point has the equation \(y = c\), where \(c\) is the y-coordinate of the point the line passes through.Since it's parallel to the x-axis and passes through \((-3, -7)\), the line will have the equation \(y = -7\).

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never intersect. They have the same slope, which means they run side by side indefinitely without meeting. In coordinate geometry, the equation of a line is often given in the form \(y = mx + c\), where \(m\) represents the slope and \(c\) the y-intercept. For two lines to be parallel, their slopes \(m_1\) and \(m_2\) must be equal, i.e., \(m_1 = m_2\).

• Equidistance: Parallel lines remain the same distance apart over their entire length.
• Horizontal example: An example of parallel lines could be two horizontal lines, like \(y = -7\) and \(y = -5\). Both lines have zero slopes and will never meet.

Understanding the concept of parallel lines is crucial when analyzing geometric figures and solving problems involving linear equations.

Horizontal Lines

Horizontal lines are unique in that they have a slope of zero. This means they do not rise or fall; rather, they extend left to right at a constant y-value. The equation of a horizontal line can be expressed simply as \(y = c\), where \(c\) is a constant that represents the y-coordinate of any point on the line.

• Slope of Zero: Because the rise over run (slope) of a horizontal line is zero, these lines stay flat and level.
• Equation Simplicity: The formula reduces the complexity involved in other linear equations as there is no x-term involved.
• Example: In the problem at hand, the line through point \((-3, -7)\) is horizontal, hence \(y = -7\).

Recognizing horizontal lines can simplify understanding certain problems in coordinate geometry and helps in quickly identifying line types in various orientations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method allows us to represent geometric shapes algebraically and form equations that describe the shape and position of those shapes on the coordinate plane.

• Points and Lines: Points are defined as ordered pairs \((x, y)\), and lines are often described using linear equations.
• Graphical Representation: This approach allows for direct visual interpretation of equations by plotting on a cartesian plane.
• Application: In our example, the coordinate geometry helps us identify the equation of the line parallel to the x-axis passing through \((-3, -7)\).

Through coordinate geometry, complex geometric concepts become easier to understand and solve as they are expressed in numerical terms. It's a fundamental tool for tackling various mathematical problems involving figures, trajectories, and spaces.