Problem 46 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,-5)\) with 0 slope

Step-by-Step Solution

Answer

The equation of the line is \ y = -5 \.

Step 1: Identify the Slope

The problem states that the slope of the line is 0. This means that the line is horizontal.

Step 2: Understand the Characteristics of a Horizontal Line

A horizontal line has the same y-value for all x-values. Therefore, the y-coordinate of every point on this line will be the same as the y-coordinate of the given point.

Step 3: Determine the Equation of the Line

Since the line is horizontal and passes through \((-1, -5)\), the y-coordinate is always -5, regardless of the x-coordinate. Thus, the equation of the line is \ y = -5 \.

Step 4: Plot the Given Point

On a coordinate plane, plot the point \((-1, -5)\). This will serve as a reference to draw the line.

Step 5: Draw the Horizontal Line

Using the point \((-1, -5)\) as a reference, draw a horizontal line that passes through this point. The y-coordinate should be -5 for all points on this line.

Key Concepts

Horizontal Line

A horizontal line is a special type of line on a coordinate plane. It runs left to right parallel to the x-axis. All points on a horizontal line have the same y-coordinate. This means that no matter what the x-value is, the y-value remains constant. For example, a line with a y-value of -5 would be written as the equation \(y = -5\). This line will cross the y-axis at -5 and extend infinitely in both directions along the x-axis.

Slope

The slope of a line indicates how steep it is. It shows the rate at which the y-coordinate of a point on the line changes with respect to the x-coordinate. A slope of 0 means there is no vertical change as the x-coordinate changes. This results in a flat, horizontal line. The formula to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For a horizontal line, \((y_2 - y_1)\) is equal to zero, so the slope is zero.

Coordinate System

A coordinate system is used to graph points, lines, and curves on a plane. It consists of two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. Points are represented as pairs of numbers (x, y), known as coordinates. The origin, where the x-axis and y-axis intersect, has coordinates (0,0). To plot a point, move x units along the x-axis and y units along the y-axis. For instance, the point \((-1, -5)\) means you move 1 unit to the left (since it's negative) and 5 units down.

Equation of a Line

The equation of a line provides a way to represent the line using algebra. One common form is the slope-intercept form, written as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) represents the y-intercept. For horizontal lines, the slope \( m \) is 0, simplifying the equation to \( y = b \). Another way to express a line's equation is through the point-slope form: \( y - y_1 = m(x - x_1) \). This is particularly useful if you have a point \((x_1, y_1)\) and the slope. For a horizontal line passing through \((-1, -5)\), the equation is simply \( y = -5 \). This shows that the y-value remains -5 for any x-value.