Problem 46 - College Algebra Essentials

Question

Give the slope and \(y\) -intercept of each line whose equation is given. Then graph the linear function. $$y=-\frac{2}{5} x+6$$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{2}{5}\) and the y-intercept is \(6\). Use these values to graph the equation.

Step 1: Identify the slope

The slope of the line can be obtained by identifying the coefficient of \(x\) in the equation of the line. In the provided equation, \(y = -\frac{2}{5}x + 6\), the slope \(m\) is \(-\frac{2}{5}\).

Step 2: Identify the y-intercept

The y-intercept can be found by identifying the constant in the equation of the line. In the given equation, \(y = -\frac{2}{5}x + 6\), the y-intercept \(b\) is \(6\).

Step 3: Graph the linear function

To plot the line, first plot the y-intercept at \(6\) on the y-axis. The slope, \(-\frac{2}{5}\), tells us that for every 5 units we move to the right on the x-axis, we move 2 units down on the y-axis. Starting from the y-intercept, move 5 units to the right and 2 units down and plot a second point. Draw the line that passes through these two points to complete the graph.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial for graphing linear functions. The slope is a measure of the steepness of a line, and it's represented by the letter 'm'. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line, often expressed as 'rise over run'.

In the equation of a line in the form of y = mx + b, 'm' represents the slope. If we consider our example, y = -\(\frac{2}{5}\)x + 6, the slope is -\(\frac{2}{5}\). This negative value indicates that the line is sloping downwards as we move from left to right. Moreover, the slope tells us that for every 5 units you step horizontally to the right on the graph, you move 2 units vertically down. Understanding this concept allows you to draw the line's inclination accurately on a graph.

Y-intercept

The y-intercept is another fundamental concept when working with linear equations. It's the point where the line crosses the y-axis, meaning it's the value of 'y' when 'x' is equal to zero.

In the standard form of a linear equation y = mx + b, 'b' represents the y-intercept. Looking at our specific function, y = -\(\frac{2}{5}\)x + 6, the y-intercept is 6. This gives us a starting point for graphing the line on a coordinate plane as it's where we put the first dot. After plotting the y-intercept on the y-axis, we can use the slope to find another point on the line. The y-intercept is essential because it provides the vertical position of the line and sets the stage for the subsequent plotting of other points.

Linear Equations

Linear equations form the backbone of graphing linear functions and are the basis for plotting lines on a coordinate plane. These equations are often presented in the classic slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The simplicity of this form provides an efficient way to visualize and sketch the behavior of lines.

In our example, y = -\(\frac{2}{5}\)x + 6, the equation is already in slope-intercept form, ready for graphing. The line represented by this equation is straightforward to plot by starting at the y-intercept (6) and following the slope to determine additional points. Remember that linear equations describe straight lines, and each line has a unique equation. Familiarizing yourself with the techniques for solving these equations and manipulating them into slope-intercept form is vital for your success in algebra and beyond.