Problem 25 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Horizontal line passing through \((2,3)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 3\).

Step 1: Identify the Slope of a Horizontal Line

A horizontal line has a slope of 0. This means the value of the dependent variable (usually y) remains constant regardless of the value of the independent variable (x).

Step 2: Determine the Equation Format for a Horizontal Line

For a horizontal line, the equation is of the form \(y = c\) where \(c\) is a constant.

Step 3: Use the Given Point to Find the Constant

Since the line passes through \((2,3)\), the y-coordinate of this point will be the constant value \c\ in the equation \(y = c\). Thus, \(c = 3\).

Step 4: Write the Final Equation

Combine the information found in the previous steps to write the equation of the line. Therefore, the equation of the line is \(y = 3\).

Key Concepts

Slope of a Line

The slope of a line tells us how steep the line is. It's an important concept in coordinate geometry.

If you think of a hill, the slope measures how much you go up or down for each step you take forward.

In coordinate geometry, the slope is calculated as the change in the vertical direction (y-coordinates) divided by the change in the horizontal direction (x-coordinates). Mathematically, it is expressed as:
\( m = \frac{{\Delta y}}{\Delta x} \).

For horizontal lines, things are special.

• The y-coordinate stays the same, making the vertical change zero.
• This results in a slope of zero.

Remember, the slope of a horizontal line is always 0.

Equation of a Line

An equation of a line represents all the points that lie on the line.

There are different forms of line equations:

• Slope-intercept form: \(y = mx + b\)
• Point-slope form: \(y - y_1 = m(x - x_1)\)
• Standard form: \(Ax + By = C\)

For horizontal lines, we use a simplified form: \(y = c\), where \(c\) is the y-coordinate that is constant.

Given a point (like (2,3)), we use the y-coordinate (3) as our constant. So, the equation of a horizontal line through (2,3) is \(y = 3\).

This means no matter what x value we have, y will always be 3.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry.

It uses a coordinate system to describe geometric figures.

Here's the basic idea:

• We use a grid formed by two perpendicular number lines, called axes.
• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.
• The point where these axes intersect is called the origin, labeled as (0,0).
• Every point in this grid can be described by an ordered pair of numbers (x, y).

A line in this grid can be described using an equation.

The equation shows the relationship between the x and y coordinates of the points on that line. Understanding this relationship helps us describe geometric shapes in an algebraic way and solve many geometric problems easily.