Problem 42 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -y=\frac{2}{3} x+\frac{11}{2} $$

Step-by-Step Solution

Answer

The slope is \(-\frac{2}{3}\) and the y-intercept is \(-\frac{11}{2}\).

Step 1: Rearrange the Equation

First, let's rearrange the given equation to put it in the slope-intercept form, which is \(y = mx + b\). The original equation is \(-y = \frac{2}{3}x + \frac{11}{2}\). Multiply or divide through by -1 to solve for \(y\): \(y = -\frac{2}{3}x - \frac{11}{2}\).

Step 2: Identify the Slope and Y-Intercept

In the slope-intercept form of a line, \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept. From the equation \(y = -\frac{2}{3}x - \frac{11}{2}\), the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is \(-\frac{11}{2}\).

Step 3: Graph the Line

To graph the equation, start by plotting the y-intercept point \((0, -\frac{11}{2})\) on the y-axis. Then, use the slope \(-\frac{2}{3}\) to find another point. From \((0, -\frac{11}{2})\), move 3 units to the right (for a positive run) and 2 units down (for a negative rise), because the slope is negative. This gives the point \((3, -\frac{15}{2})\). Draw a line through these points to represent the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental concept in algebra. It's a way to rewrite linear equations so that the key components of a line are easily recognizable. The slope-intercept form of a linear equation is expressed as:
\[ y = mx + b \]
Here, \( m \) represents the slope of the line, while \( b \) stands for the y-intercept. This formula is particularly useful because it allows you to quickly graph a line. By identifying \( m \) and \( b \), we can easily understand how the line behaves. The slope \( m \) tells us how steep the line is and whether it slopes upwards or downwards. Meanwhile, the y-intercept \( b \) shows us where the line crosses the y-axis. To rearrange any linear equation into this form, you simply need to solve for \( y \). Once in this form, everything about the line becomes much clearer and manageable.

Graphing Linear Equations

When it comes to graphing linear equations, the slope-intercept form serves as a handy tool. Graphing starts with plotting the y-intercept point on the y-axis. This point, represented as \((0, b)\), anchors your line on the graph.
Next, using the slope \( m \), you determine the direction and steepness of the line. The slope \( -\frac{2}{3} \) in this case means for every 3 units you move to the right, you must move 2 units down. A slope of \(-\frac{2}{3}\) suggests a negative, or downward trend from left to right.
After plotting the initial y-intercept, you apply the slope to find your next point. Continue this process at least once to ensure accuracy and then draw a straight line through the points. This line represents all the solutions to the linear equation that you started with.

Identifying Slope and Y-Intercept

Recognizing the slope and y-intercept in an equation is crucial for understanding and manipulating linear equations. In a slope-intercept form equation \( y = mx + b \):

• The slope (m) indicates the vertical change for each unit of horizontal change. It can be a whole number, fraction, positive, or negative, which dictates the line's tilt.
• The y-intercept (b) denotes the point on the graph where the line meets the y-axis. This is where \( x = 0 \).

Given an equation, like \( y = -\frac{2}{3}x - \frac{11}{2} \), identifying these elements becomes straightforward. The slope \( m = -\frac{2}{3} \) implies the line will decline as it moves across the graph, while the y-intercept \( b = -\frac{11}{2} \) pinpoints the exact spot the line will touch the y-axis. Identifying these components not only helps in graphing the line but also enables predictions about how the line behaves without drawing it.