Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=-\frac{2}{3} x+4$$

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{2}{3} x+4\) is a line that intercepts the y-axis at the point (0, 4) and has a slope of -2/3.

Step 1: Identify the slope and y-intercept

In the equation \(y=-\frac{2}{3} x+4\), the slope is -2/3 and the y-intercept is 4. This can be observed as our equation is in the slope-intercept form \(y = mx + c\), where \(m\) represents the slope and \(c\) represents the y-intercept.

Step 2: Plot the y-intercept

Start by plotting the y-intercept on the graph, which is the point (0, 4). It is the point where the line crosses the y-axis.

Step 3: Use the slope to find the next point

The slope -2/3 means that for every 3 units we move to the right along the x-axis, we move 2 units down along the y-axis. So from the y-intercept (0,4), move 3 units to the right and 2 units down which gives the point (3, 2). Plot this point.

Step 4: Draw the line

Use the two points to draw the line representing the solution to the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is pivotal for understanding and graphing linear equations. It is commonly expressed as \( y = mx + c \). Here, \( m \) denotes the slope, while \( c \) stands for the y-intercept. This form is particularly convenient because it allows you to quickly identify both the slope and the y-intercept from any linear equation. This makes graphing simple and efficient, especially when you need to plot the line.

When a linear equation is in the slope-intercept form, graphing becomes a straightforward process. You begin by marking the y-intercept on the graph and then use the slope to determine the direction and steepness of the line.

For example, the equation \( y = -\frac{2}{3}x + 4 \) tells you that the slope is \(-\frac{2}{3}\), and the y-intercept is 4. You can directly use these values to start your graphing work.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that helps visualize relationships between variables. It involves plotting the equation on a coordinate plane to see where the line connects various points.

Start by identifying the y-intercept, which is where the line crosses the y-axis. For the equation \( y = -\frac{2}{3}x + 4 \), the y-intercept is 4, so you plot the point \( (0, 4) \) on the graph. Next, deploy the slope to determine the direction and how far other points lie from this intercept.

For example, with a slope of \(-\frac{2}{3}\), you move down 2 units and 3 units to the right from the y-intercept to find another point. Plotting these points gives you a clear visualization of the line, which can then be drawn by connecting the dots smoothly with a straightedge.

Slope

In linear equations, the slope is a measure that describes the steepness and direction of the line. It is represented by the letter \( m \) in the equation \( y = mx + c \). The slope can be positive, negative, zero, or undefined:

• Positive slope: The line rises from left to right.
• Negative slope: The line falls from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

The slope \(-\frac{2}{3}\) in the equation \( y = -\frac{2}{3}x + 4 \) means the line moves down 2 units for every 3 units it moves to the right. Calculating and understanding slope helps you graph lines accurately and understand how variables in the equation interact.

Y-Intercept

The y-intercept is a crucial concept in graphing, indicating where the line crosses the y-axis. In the slope-intercept form \( y = mx + c \), \( c \) is the y-intercept. The y-intercept tells you the value of \( y \) when \( x \) is 0. This gives you a starting point for graphing the line.

For the equation \( y = -\frac{2}{3}x + 4 \), the y-intercept is 4. This means that the first point you plot is \( (0, 4) \) on the y-axis.

After plotting the y-intercept, you can rely on the slope to find additional points, ensuring the line is graphed accurately. The y-intercept provides a straightforward way to anchor your line to the graph, making it an indispensable part of the graphing process.