Problem 9 - Elementary Algebra

Question

Graph the equations. $$ y=-\frac{8}{3} x+4 $$

Step-by-Step Solution

Answer

Question: Graph the linear equation y = -8/3x + 4. Answer: Follow the steps below to graph the given linear equation: 1. Identify the slope and the y-intercept: m = -8/3 and b = 4 2. Plot the y-intercept on the graph: (0, 4) 3. Use the slope to find more points: One point to the right -> (3, -4); one point to the left -> (-3, 12) 4. Connect the points (0, 4), (3, -4), and (-3, 12) with a straight line to form the graph of the equation y = -8/3x + 4.

Step 1: 1. Identify the slope and the y-intercept

Looking at the given equation: $$ y = -\frac{8}{3} x + 4 $$ We can identify the slope (m) and y-intercept (b) as follows: $$ m = -\frac{8}{3}, \quad b = 4 $$

Step 2: 2. Plot the y-intercept

The y-intercept is where the line crosses the y-axis, which means x=0. So we need to plot the point (0, 4).

Step 3: 3. Use the slope to find more points

The slope of the line is: $$ m = -\frac{8}{3} $$ This means that for every 3 units we move to the right on the x-axis, we need to go down by 8 units on the y-axis. So starting from the y-intercept (0, 4), we take 3 steps on the x-axis to the right and 8 steps down on the y-axis, which leads us to the point (3, -4). Similarly, if we move 3 units to the left of the y-intercept (0, 4) on the x-axis, we go up by 8 units on the y-axis. This brings us to the point (-3, 12).

Step 4: 4. Connect the points to form the line

Now that we have three points, we can connect them to form the line representing the graph of the given linear equation: (0, 4) -> (-3, 12) -> (3, -4) By connecting these points with a straight line, we have successfully graphed the equation: $$ y = -\frac{8}{3} x + 4 $$

Key Concepts

Slope-Intercept Form

Understanding slope-intercept form is central to grasping the basics of linear equations. In its simplest terms, slope-intercept form is written as \( y = mx + b \), where \( m \) stands for the slope of the line and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

When graphing linear equations using the slope-intercept form, one can easily determine the initial point where the line will cross the y-axis (the y-intercept) and how steep the line will be (the slope). For instance, in the given exercise, the equation \( y = -\frac{8}{3}x + 4 \) shows a slope \( m = -\frac{8}{3} \) and a y-intercept \( b = 4 \). This notation streamlines the process of graphing as it gives a clear starting point and a consistent method to follow for plotting the line on a coordinate plane.

Y-Intercept

The y-intercept of a line is where the graph of the equation crosses the y-axis. This means it is the point where the value of \( x \) is zero. Focusing on the y-intercept allows for a solid starting place for graphing the equation. In our exercise, the y-intercept, represented by \( b \) in the slope-intercept form, is 4. Therefore, one would begin plotting the graph at point \( (0, 4) \).

This value provides an immediate visual reference point on the y-axis, and from there, we can use the slope to determine the direction and the steepness of the line.

Slope of a Line

The slope of a line is a numerical value that represents the 'steepness' or the incline of the line. A positive slope indicates that the line rises as one moves from left to right, whereas a negative slope signifies that the line falls. The slope of a line is often represented by the letter \( m \) in the equation of a line.

In the slope-intercept form \( y = mx + b \), \( m \) tells us how many units up or down we move on the y-axis for every single unit we move to the right on the x-axis. In our exercise, the slope is \( -\frac{8}{3} \), indicating a downward movement of 8 units for every 3 units we move to the right. Slope provides direction and slant to the graph and is crucial in visually interpreting the rate of change expressed by the linear equation.

Plotting Points

Plotting points is a fundamental skill needed to graph any equation. It involves drawing exact spots on a graph using pairs of x and y coordinates. Once you have your slope and y-intercept from an equation in slope-intercept form, plotting points becomes a systematic process.

As shown in the exercise, you start with the y-intercept (0, 4) and then use the slope to find other points on the line. To plot points using a slope of \( -\frac{8}{3} \), from (0, 4), you'd move 3 units to the right (positive direction along the x-axis) and 8 units down (negative direction along the y-axis) to find the next point at (3, -4). Likewise, moving in the opposite direction, that is 3 units to the left and 8 units up, we find another point at (-3, 12). Precise plotting ensures the line graphed is an accurate representation of the equation.