Problem 46 - College Algebra

Question

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

Step-by-Step Solution

Answer

The slope of the given line is \(\frac{2}{5}\) and the y-intercept is 6.

Step 1: Identify Slope and Y-Intercept

From the given equation \(y--\frac{2}{5} x+6\), we see that the slope \(m=\frac{2}{5}\) and the y-intercept \(b=6\).

Step 2: Graph the Intercept

To start graphing the line, plot the y-intercept on the y-axis. This is the point \(0,6\).

Step 3: Use the Slope to Plot the Line

The slope is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Since the slope is \(\frac{2}{5}\), for every 5 units we move to the right (positive direction) along the x-axis, we move 2 units upwards (positive direction) along the y-axis. This gives us another point on the line. Using these two points, draw the line.

Key Concepts

Slope of a Line

When we talk about the slope of a line in algebra, we're referring to a number that describes both the direction and the steepness of the line. The slope is usually represented by the letter 'm'. Imagine you’re climbing a hill; the slope tells you how steep the hill is. The steeper the hill, the larger the slope number.

The slope is calculated as the ratio of the 'rise' (the vertical change) to the 'run' (the horizontal change) between any two points on the line. If we have two points, say \( (x_1,y_1) \) and \( (x_2,y_2) \), the slope can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). A positive slope means the line rises as it goes from left to right, while a negative slope means the line falls. If the slope is zero, the line is horizontal, and if the slope is undefined (the run is zero), the line is vertical.

In our exercise's context, \( m=\frac{2}{5} \) means for every five units we move to the right (positive run), the line goes up (positive rise) by two units, creating a line that slants upwards as it moves from left to right.

Y-Intercept

Now, let's focus on the y-intercept. This is where our line crosses the y-axis. The y-intercept is typically represented by the letter 'b' in the slope-intercept form of a line's equation, which looks like \( y = mx + b \). It's the point where the value of x is zero. Understanding the y-intercept is crucial because it gives us a starting point to graph our line.

In simpler terms, if you’re standing at the point where a hill starts to incline, that's your y-intercept on the graph of the hill's profile. In the exercise, the y-intercept is \(6\), which means the line crosses the y-axis at the point \( (0, 6) \). This gives us a fixed point from which we can begin to plot the rest of the line using the slope.

Plotting Points on a Graph

Plotting points on a graph is like drawing a map that shows the locations of different landmarks. To plot a point, we need an x-coordinate and a y-coordinate which tell us the horizontal and vertical positions of the point, respectively. By plotting multiple points and connecting them, we create a visual representation of our line.

Let's apply this to the exercise. With the y-intercept \( (0, 6) \) already plotted, we use the slope \( \frac{2}{5} \) to determine the next point. Starting from \( (0, 6) \), we move 5 units to the right on the x-axis because our slope 'run' is 5. Then, we move 2 units up on the y-axis because our slope 'rise' is 2. We've plotted our second point. All that's left is to connect these dots with a straight line, and there you have it: the graph of the line with slope \( \frac{2}{5} \) and y-intercept 6.