Problem 31 - Elementary Algebra

Question

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=-\frac{5}{9}\) and \(b=-\frac{1}{2}\)

Step-by-Step Solution

Answer

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

Step 1: Identify Given Values

We are given the slope \(m\) and the \(y\)-intercept \(b\). Specifically, \(m = -\frac{5}{9}\) and \(b = -\frac{1}{2}\).

Step 2: Recall Slope-Intercept Form

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

Step 3: Substitute Values into Formula

Replace \(m\) with \(-\frac{5}{9}\) and \(b\) with \(-\frac{1}{2}\) in the formula: \[ y = -\frac{5}{9}x - \frac{1}{2} \]

Step 4: Write the Final Equation

The equation of the line in slope-intercept form is \[ y = -\frac{5}{9}x - \frac{1}{2} \]

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression representing all points that lie on the line. The most common forms for these equations are the slope-intercept form, the point-slope form, and the standard form. For this exercise, we focus on the slope-intercept form.

In the slope-intercept form, a line’s equation is written as:

• Slope-intercept form: \(y = mx + b\)

The equation gives you direct information about the line’s slope and the location where it intersects the y-axis, known as the y-intercept. In this format, the linear equation is straightforward because it directly uses the slope \(m\) and the y-intercept \(b\) to describe the line's unique characteristics.

Slope

The slope of a line is a measure of its steepness and direction. It describes how much the line rises or falls as you move along it from left to right. Mathematically, the slope \(m\) is the ratio of the change in y (vertical change) to the change in x (horizontal change).

• Slope equation: \(m = \frac{\Delta y}{\Delta x}\)
• A positive slope means the line ascends from left to right.
• A negative slope indicates the line descends as you move from left to right.

In our given problem, the slope \(-\frac{5}{9}\) means that for every 9 units we move to the right on the x-axis, the line falls 5 units on the y-axis. This negative value shows that our line is sloping downwards as it progresses from left to right.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis. It provides the exact y-coordinate when the x-coordinate is zero. The y-intercept is represented by the letter \(b\) in the slope-intercept equation \(y = mx + b\).

• The y-intercept shows the value of \(y\) when \(x = 0\).
• It is an essential value which indicates where the line starts on the vertical y-axis.

In the context of our problem, the y-intercept is \(-\frac{1}{2}\). This means that the line will intersect the y-axis at the point (0, -1/2). It's the constant term in the equation and sets the starting height of the line on the graph. Knowing the y-intercept helps in quickly visualizing and sketching the line on a graph.