Problem 42 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Slope: -2/3, Y-intercept: -5.5. Graph the line using these points.

Step 1: Rearrange the Equation

We start with the given equation:\[-y = \frac{2}{3}x + \frac{11}{2}\]First, multiply the entire equation by -1 to make the coefficient of y positive:\[y = -\frac{2}{3}x - \frac{11}{2}\].

Step 2: Identify the Slope and Y-intercept

Now that the equation is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, we can identify:- The slope \(m\) is \(-\frac{2}{3}\).- The y-intercept \(b\) is \(-\frac{11}{2}\), or -5.5.

Step 3: Graph the Line

To graph the line, start by plotting the y-intercept at \( (0, -5.5) \) on the y-axis. From this point, use the slope of \(-\frac{2}{3}\) to find another point on the line:- Since the slope is \(-\frac{2}{3}\), it means that for every 3 units you move to the right along the x-axis, you move 2 units down along the y-axis.- From \( (0, -5.5) \), move 3 units to the right to \( (3, -5.5) \), then move 2 units down to \( (3, -7.5) \).- Draw a line through these points to complete the graph.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is an important skill in algebra. A linear equation describes a straight line on a coordinate plane. The general form of a linear equation in two variables is the slope-intercept form, which is written as \[y = mx + b\]. This form makes it simple to graph the line by providing two key features right away:

• the slope (\(m\)), which tells us the steepness of the line,
• the y-intercept (\(b\)), which is the point where the line crosses the y-axis.

To graph a linear equation:
1. Start by identifying the y-intercept and plotting it on the y-axis.
2. Use the slope to determine the direction and steepness of the line.

For example, if the equation is \[-y = \frac{2}{3}x + \frac{11}{2}\], we first rearrange it to the slope-intercept form: \[y = -\frac{2}{3}x - \frac{11}{2}\]. Then, plot the y-intercept at (0, -5.5) and use the slope \(-\frac{2}{3}\) to find another point like (3, -7.5). Draw a straight line through these points to create the graph of the equation.

Slope of a Line

The slope of a line is a measure of its steepness and the direction it goes. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line.
Mathematically, if a line passes through two points \((x_1, y_1)\) and \((x_2, y_2)\), its slope \(m\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In the slope-intercept form of a linear equation, \(y = mx + b\), \(m\) represents the slope.
A positive slope means the line goes up as it moves from left to right, while a negative slope means the line goes down. A zero slope means the line is horizontal, and an undefined slope means the line is vertical.
In our specific equation \[y = -\frac{2}{3}x - \frac{11}{2}\], the slope \(-\frac{2}{3}\) means that for every 3 units you move to the right, you go 2 units down, giving it a negative direction. This makes the graph of the line descend as we move from left to right.

Y-Intercept

The y-intercept is a vital part of the slope-intercept form of a linear equation. It is the point where the line crosses the y-axis, and it can be easily found in the equation \(y = mx + b\) as the constant \(b\).
When you have this equation, the y-intercept tells you the value of \(y\) when \(x\) is 0. It provides a starting point for graphing the line by allowing us to quickly place one fixed point on the graph.

In the example with the equation \(-y = \frac{2}{3}x + \frac{11}{2}\), after rearranging to \(y = -\frac{2}{3}x - \frac{11}{2}\), the y-intercept is \(-\frac{11}{2}\), or -5.5 in decimal form. This means the line passes through the point \((0, -5.5)\) on the y-axis.
Knowing the y-intercept along with the slope allows for the complete graphing of the linear equation, by starting at the y-intercept and using the slope to determine the other points on the line.