Problem 33 - Introductory Algebra for College Students

Question

Graph each linear equation using the slope and y-intercept. $$y=\frac{2}{3} x-5$$

Step-by-Step Solution

Answer

The graph of the equation \(y=\frac{2}{3}x-5\) is a straight line passing through points (0, -5) and (3, -3).

Step 1: Identify the Slope (m) and Y-intercept (b)

From the equation, it can be derived that the slope (m) is \(\frac{2}{3}\) and y-intercept (b) is -5.

Step 2: Plot the Y-Intercept on the Graph

The y-intercept is the point where the line crosses the y-axis. Here, \(b = -5\) so the point on y-axis is (0, -5). Begin by plotting this point on y-axis.

Step 3: Use the Slope to Find Another Point

The slope (\(m\)) is \(\frac{2}{3}\). It means that for each increase of 3 units to the right along the x-axis (run), the y-value (rise) increases by 2 units. From the y-intercept (0,-5), count 3 units to the right and 2 units up to get a second point on the line (3, -3).

Step 4: Draw the Line Through the Points

Now that you have at least two points, you can draw a straight line through these points to graph the equation. This line represents all the solutions for the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a standard way of writing linear equations, which allows you to quickly identify key features of the line. In mathematics, it is represented by the equation \( y = mx + b \).

Here, \( m \) is the slope of the line, which indicates how steep the line is. The slope tells you how much the line rises vertically for every unit it moves horizontally.

Meanwhile, \( b \) is the y-intercept of the line, showing where the line crosses the y-axis. It is useful for easily sketching the graph of the line by starting from the y-intercept and using the slope to find other points.

This form is widely used in both mathematics and real-world applications because it provides clear and essential information about the linear relationship.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis on a graph. It is an important feature because it gives you a starting point for drawing the line.

In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). It shows the value of \( y \) when \( x = 0 \).

For the equation \( y = \frac{2}{3}x - 5 \), the y-intercept \( b \) is \(-5\). This means the line crosses the y-axis at the point \((0, -5)\). Plotting the y-intercept is the first step in graphing the line, and from there, we can use the slope to determine other points.

Plotting Points

Plotting points is a crucial step in graphing linear equations. Once you know the y-intercept, you can plot it on the graph.

From the equation \( y = \frac{2}{3}x - 5 \), you first plot the y-intercept \((0, -5)\).

Using the slope, find another point. Start at the y-intercept, then apply the slope \(\frac{2}{3}\). This means moving 3 units to the right (along the x-axis) and 2 units up (along the y-axis).

Now you have a second point, \((3, -3)\), which can be plotted. With two points: the y-intercept and the point found through the slope, you can draw a line through them to represent the equation on the graph.

Slope Calculation

Calculating slope accurately is key to understanding how steep a line is on a graph.

The slope \( m \) is expressed as the ratio of the rise (change in y) to the run (change in x).

For the equation \( y = \frac{2}{3}x - 5 \), the slope \( \frac{2}{3} \) means that for every 3 units you move to the right (along the x-axis), the line moves 2 units up (along the y-axis).

This positive slope indicates an upward trend on the graph. Knowing how to calculate and interpret the slope can help you determine the direction and steepness of the line, which is essential for graphing accurate representations of linear equations.