Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$2 x+9 y=0$$

Step-by-Step Solution

Answer

The slope \(m\) of the line is -2/9 and the y-intercept \(b\) is 0.

Step 1: Transforming the equation into slope-intercept form

First, isolate \(y\) to put the equation in slope-intercept form. Solve for \(y\) in the equation \(2x + 9y = 0\). This involves subtracting \(2x\) from both sides and then dividing by 9. The equation then becomes \(y = -\frac{2}{9}x + 0\).

Step 2: Identifying the slope

In a slope-intercept form equation \(y = mx + b\), the coefficient of \(x\) is the slope. Therefore in our equation \(y = -\frac{2}{9}x + 0\), the slope or \(m\) is -\frac{2}{9}.

Step 3: Identifying the y-intercept

Similarly, in a slope-intercept form equation \(y = mx + b\), \(b\) is the y-intercept. Therefore in our equation \(y = -\frac{2}{9}x + 0\), the y-intercept or \(b\) is 0.

Key Concepts

Linear Equations

Linear equations are the simplest form of equations you'll encounter in algebra. They describe a relationship between two variables with a constant rate of change. That's why when you graph a linear equation, you always get a straight line. The general form of a linear equation in two variables, x and y, is expressed as \( ax + by = c \).

To solve linear equations and make them easier to comprehend, we often rearrange them into what is known as the slope-intercept form, which looks like \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) denotes the y-intercept. This form is super useful because it gives you the slope and y-intercept directly, which helps tremendously in graphing the line. In our textbook exercise, we began with \( 2x + 9y = 0 \) and rearranged it to \( y = -\frac{2}{9}x \), which simplifies to slope-intercept form with a slope of \( -\frac{2}{9} \) and a y-intercept of 0.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's the 'm' in the slope-intercept form equation \( y = mx + b \). A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. If the slope is zero, the line is horizontal, and if it's undefined (division by zero), the line is vertical.

In the example given, we identified the slope by rearranging the equation to the slope-intercept form, resulting in \( y = -\frac{2}{9}x \). Here, the slope is \( -\frac{2}{9} \), indicating the line falls as it goes from left to right, and for every nine units you move horizontally, the line moves down two units vertically. Understanding slope is critical because it describes the rate at which one variable changes with respect to the other, which can be applied in various real-world scenarios, such as calculating inclines or tracking changes over time.

Y-Intercept

The y-intercept is where the graph of an equation intersects the y-axis. This point represents the value of y when x is zero. In the slope-intercept form \( y = mx + b \) of a linear equation, \( b \) is the y-intercept. It's essential in understanding the starting point of the line on a graph.

In our problem, after transforming the equation into slope-intercept form, we found the y-intercept to be 0, indicated by the equation \( y = -\frac{2}{9}x + 0 \). This tells us the line goes through the origin, which is the point (0,0). In practice, if you can identify the y-intercept, you already have a starting point from which you can use the slope to find other points on the line, making graphing the equation a piece of cake.