Problem 24 - 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 \(b=4\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{5}{9}x + 4 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line equation is given by \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. This form makes it easy to identify the slope and the y-intercept directly from the equation.

Step 2: Insert Given Values

Substitute the given slope \( m = \frac{5}{9} \) and y-intercept \( b = 4 \) into the slope-intercept equation. This gives us the equation: \( y = \frac{5}{9}x + 4 \).

Step 3: Write the Equation in Slope-Intercept Form

After substituting the values, ensure that the equation \( y = \frac{5}{9}x + 4 \) is in the slope-intercept form where \( m \) and \( b \) are clearly identifiable.

Key Concepts

Slope

The slope of a line, often represented by the letter \( m \), measures how steep the line is. It indicates the rate at which the \( y \)-value of a point on the line changes with respect to changes in the \( x \)-value. The slope is calculated as the "rise" over the "run," which means:

• "Rise" is the vertical change between two points on the line.
• "Run" is the horizontal change between these two points.

For example, if the slope \( m \) is \( \frac{5}{9} \), it tells you that for every 9 units you move to the right, you will move 5 units up. This gives you a clear idea of how to move from one point to another along the line. A positive slope means the line inclines upwards as you move from left to right, while a negative slope means it declines. Understanding the slope can help in predicting how the line behaves on a graph.

Y-Intercept

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. This point is represented as \( (0, b) \) where \( b \) is the y-intercept. When a line intersects with the \( y \)-axis, the \( x \)-coordinate is always zero. The \( y \)-intercept provides a starting point for drawing the line on a graph.

• In our example, the \( y \)-intercept is \( 4 \).
• This means the line intersects the \( y \)-axis at the point \( (0, 4) \).

Y-intercepts are easy to find because they just show where the line begins as you start graphing, and they help set the initial position of the line on the graph. Recognizing the \( y \)-intercept through the equation and graphing it gives you a firm foundation on which to build understanding of how lines behave visually.

Equation of a Line

The equation of a line in slope-intercept form is written as \( y = mx + b \). This format is very useful because it directly provides the slope \( m \) and the y-intercept \( b \). Here is how each part contributes:

• The \( m \) in the equation shows the slope, or the tilt of the line.
• The \( b \) stands for the y-intercept, where the line cuts through the \( y \)-axis.

For instance, if the equation is \( y = \frac{5}{9}x + 4 \), it reveals automatically that:

• The slope \( m \) is \( \frac{5}{9} \).
• The y-intercept \( b \) is \( 4 \).

This form is particularly helpful when you are graphing the line since it tells you exactly where to start and how to continue drawing the line based on the slope. Knowing how to quickly write and interpret this form, using the slope and \( y \)-intercept, makes understanding linear equations easier.