Problem 28 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Horizontal line passing through \((4,-3)\)

Step-by-Step Solution

Answer

y = -3

Step 1: Understanding the Problem

We need to write the equation of a line that is horizontal and passes through the point \(4, -3\). A horizontal line has a constant y-value across all x-values.

Step 2: Identify the Form of a Horizontal Line Equation

The general equation of a horizontal line is \(y = c\) where \(c\) is the y-coordinate of every point on the line.

Step 3: Substitute the Given Point

Since the given line passes through the point \(4, -3\), the y-coordinate (c) is \(-3\). Therefore, the equation is \(y = -3\).

Key Concepts

horizontal line

A horizontal line is a straight line that extends left to right and remains at a constant y-value. It doesn't tilt up or down at any degree. This means that no matter what the x-coordinate is, the y-coordinate stays the same. For instance, if a line is horizontal and passes through the point \(4, -3\), the y-value is always \-3\. Hence, the equation of this horizontal line is simply \(y = -3\). This concept is easy once you remember that horizontal lines have no slope, meaning the change in y is zero across all x-coordinates.

point-slope form

The point-slope form of a line allows us to write the equation of a line given its slope and a point it passes through. The formula is written as: \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope, and \( (x_1, y_1) \) represents the point on the line. However, for a horizontal line, the slope \(m\) is zero. This simplifies things since no matter the x-value, the y-value remains constant, as highlighted in our previous section. Since a horizontal line doesn't change in the y-direction, the point-slope form essentially doesn't apply here specifically other than emphasizing the unchanging y-coordinate.

coordinate geometry

Coordinate geometry uses algebraic techniques to study geometry, translating geometric shapes into algebraic equations. When dealing with lines, we often use different forms to represent their equations: * **Slope-intercept form (y = mx + b)**: perfect for non-horizontal or non-vertical lines. * **Point-slope form (y - y_1 = m(x - x_1))**: useful when we know a point and the slope. * **Horizontal line (y = c)**: straightforward, where \(c\) is a constant. For example, a horizontal line passing through point \(4, -3\) translates algebraically to \(y = -3\). Coordinate geometry provides these various forms for convenience, ensuring we can describe different types of lines effectively.