Problem 24 - 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. $$ m=\frac{5}{9} \text { and } b=4 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Slope-Intercept Form Equation

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept of the line. In this exercise, we're asked to find the equation of the line using the given slope and y-intercept.

Step 2: Substitute the Given Values

Substitute the given value of the slope \(m = \frac{5}{9}\) and the y-intercept \(b = 4\) into the slope-intercept form equation. Thus, the equation becomes \(y = \frac{5}{9}x + 4\).

Key Concepts

Linear Equations

A linear equation is a mathematical expression that represents a straight line when plotted on a graph. The general form of a linear equation is written as \(y = mx + b\), where:

• \(y\) represents the dependent variable, which depends on the value of \(x\), the independent variable.
• \(m\), the slope, indicates how steep the line is.
• \(b\), the y-intercept, is where the line crosses the \(y\)-axis.

In linear equations, the relationship between \(y\) and \(x\) is proportional, meaning as \(x\) increases or decreases, \(y\) changes in a consistent, linear manner. This is why these equations are called "linear," because when graphed, they form a line. Understanding linear equations is fundamental because they form the foundation of algebra and are used extensively in fields like physics, engineering, and economics.

Slope

The slope of a line, denoted as \(m\) in the slope-intercept form \(y = mx + b\), represents the rate at which \(y\) changes with respect to \(x\). In simple terms, the slope tells you how steep the line is:

• A larger slope value means the line is steeper.
• A positive slope indicates the line ascends from left to right, showing a positive correlation.
• A negative slope means the line descends from left to right, showing a negative correlation.
• A zero slope signifies a horizontal line, where \(y\) doesn’t change as \(x\) changes.

Mathematically, slope is calculated as the "rise" over the "run," or the change in \(y\) over the change in \(x\). For the exercise given, the slope \(m=\frac{5}{9}\) implies that for every 9 units \(x\) increases, \(y\) increases by 5 units. This ratio is crucial in determining how sharply the graph of the equation inclines or declines.

Y-Intercept

The y-intercept (\(b\) in the equation \(y = mx + b\)) is a key element in understanding linear equations. It tells us where the line intersects the \(y\)-axis. In other words, it's the value of \(y\) when \(x\) is 0.The y-intercept is essential because:

• It provides a starting point for the line's graph.
• In a real-world context, it often represents an initial value or starting condition.

For the exercise problem, the y-intercept is \(b = 4\). This means that at \(x = 0\), the value of \(y\) is 4. Knowing the y-intercept helps us quickly sketch the line by plotting this initial point, which serves as a foundation to draw the rest of the line according to the slope value. This concept is widely used in scenarios like predicting future events or interpreting initial conditions in data analysis.