Problem 31 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-\frac{5}{9} \text { and } b=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The equation is \( y = -\frac{5}{9}x -\frac{1}{2} \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify Given Values

Here, the slope \( m \) is \(-\frac{5}{9}\) and the y-intercept \( b \) is \(-\frac{1}{2}\).

Step 3: Substitute Values into the Equation

Substitute \( m = -\frac{5}{9} \) and \( b = -\frac{1}{2} \) into the slope-intercept form equation to get \( y = -\frac{5}{9}x -\frac{1}{2} \).

Step 4: Conclusion

The equation of the line with the given slope and y-intercept is \( y = -\frac{5}{9}x -\frac{1}{2} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a line. It is written as \( y = mx + b \). Here, each part of the formula has a specific meaning:
\( y \) represents the dependent variable, which changes in relation to \( x \), the independent variable. The relationship between \( x \) and \( y \) is determined by two key components: the slope and the y-intercept.
In everyday terms, you can think of this form as a description of a line. By knowing just the slope and y-intercept, you can precisely draw the line on a graph. This way, it becomes an essential tool in algebra for easily communicating and working with linear equations.

• \( m \) is the slope and tells us how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Understanding how to use this format is crucial for solving problems that involve linear relationships.

Slope of a Line

The slope of a line, denoted as \( m \), is a measure of how steep a line is. It shows the rate at which \( y \) changes with respect to changes in \( x \). Imagine walking up a hill, the steeper the hill, the bigger the slope.
The slope is calculated as the 'rise' over the 'run'. This means:\[ m = \frac{\text{change in } y}{\text{change in } x} \]
The slope can tell us if a line is going upwards or downwards:

• If \( m \) is positive, the line goes up as we move from left to right.
• If \( m \) is negative, the line goes down as we move from left to right.

A larger slope means a steeper incline or decline. For example, a slope of \(-\frac{5}{9}\) tells us that for every 9 units we move horizontally, the line drops by 5 units vertically. This kind of slope helps us understand the behavior of our line in a coordinate plane and how different points on the line relate to each other.

Y-intercept

The y-intercept is the point where a line crosses the y-axis on a graph. It's an important concept because it shows the starting point of a line when \( x \) is zero.
In the equation of a line in slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept. This value can quickly be found by setting \( x = 0 \) in the equation. The point where \( x = 0 \) and \( y \) equals \( b \) is the y-intercept.
Understanding the y-intercept helps in graphing and interpreting linear equations:

• If \( b \) is positive, the line crosses the y-axis above the origin.
• If \( b \) is negative, the line crosses below the origin.

For the equation \( y = -\frac{5}{9}x -\frac{1}{2} \), the y-intercept is \(-\frac{1}{2}\). This means the line crosses the y-axis at the point \((0, -\frac{1}{2})\). Knowing where the line starts is essential for understanding how it fits within the overall graph.