Problem 68 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=-\frac{3}{4} x+\frac{1}{5}$$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{3}{4}\).

Step 1: Choose an Easy X-Value to Find the First Point

To find a point on the line, choose an easy value for \( x \). A common choice is \( x = 0 \). Substitute \( x = 0 \) into the equation: \[ y = -\frac{3}{4}(0) + \frac{1}{5} = \frac{1}{5} \]Hence, the first point is \((0, \frac{1}{5})\).

Step 2: Choose Another X-Value to Find the Second Point

Choose another simple value for \( x \), like \( x = 4 \), to compute a different point on the line. Substitute \( x = 4 \) into the equation: \[ y = -\frac{3}{4}(4) + \frac{1}{5} = -3 + \frac{1}{5} = -\frac{15}{5} + \frac{1}{5} = -\frac{14}{5} \]So, the second point is \((4, -\frac{14}{5})\).

Step 3: Use the Coordinates to Find the Slope

With two points \((x_1, y_1) = (0, \frac{1}{5})\) and \((x_2, y_2) = (4, -\frac{14}{5})\), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the known values:\[ m = \frac{-\frac{14}{5} - \frac{1}{5}}{4 - 0} = \frac{-\frac{14}{5} - \frac{1}{5}}{4} = \frac{-\frac{15}{5}}{4} = \frac{-3}{4} \]Therefore, the slope of the line is \(-\frac{3}{4}\).

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and direction. When dealing with a linear equation, the slope is denoted by the letter \( m \). To find the slope, you need two points on the line. The slope is calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula simply represents the rise over run—how much the line rises vertically compared to how much it runs horizontally. In our exercise, two points are identified: \((0, \frac{1}{5})\) and \((4, -\frac{14}{5})\). By substituting these into the slope formula, we find the slope to be \(-\frac{3}{4}\). From this value, the negative sign implies the line is decreasing as it moves from left to right. Meanwhile, the fraction \(\frac{3}{4}\) means that for every 4 units the line moves horizontally, it drops 3 units vertically.

Graphing Lines

When graphing lines, one effective method is using the slope-intercept form of a linear equation, given as \( y = mx + b \). In this format:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

In the original equation \( y = -\frac{3}{4}x + \frac{1}{5} \), the slope \( m \) is \(-\frac{3}{4}\) and the y-intercept \( b \) is \(\frac{1}{5}\). To start graphing, plot the y-intercept, \((0, \frac{1}{5})\). From this point, use the slope to find the next point: move 4 units right and 3 units down because the slope is negative. Connecting these points with a straight line gives the graph of the equation. Graphing visually presents data and helps in analyzing trends or relationships between the variables.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. Points are placed on the plane using coordinates \((x, y)\), which offer a precise location. This exercise uses the Cartesian coordinate plane, a typical system made up of horizontal and vertical lines known as axes. In coordinate geometry, the position of a point is defined using two numbers (coordinates), reflecting its distance from the axes. Using equations, we understand lines, curves, and their positions on this grid. For example, the equation \( y = -\frac{3}{4}x + \frac{1}{5} \) describes a straight line. Understanding this concept helps recognize how equations relate to their graphic representations, showing intersections and relations between lines—skills vital in diverse fields including physics, engineering, and computer graphics.