Problem 22 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Slope \(\frac{2}{5} ; \quad y\) -intercept 4

Step-by-Step Solution

Answer

The line equation is \( y = \frac{2}{5}x + 4 \).

Step 1: Identifying Line Equation Formula

A line's equation in slope-intercept form is given by the formula \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substituting the Slope Value

We know the slope \( m = \frac{2}{5} \). Substitute \( m \) into the line equation formula to get \( y = \frac{2}{5}x + b \).

Step 3: Substituting the y-intercept Value

We are given that the y-intercept \( b = 4 \). Substitute \( b \) into the equation to get \( y = \frac{2}{5}x + 4 \).

Step 4: Verifying the Equation

The line equation \( y = \frac{2}{5}x + 4 \) satisfies the given conditions: a slope of \( \frac{2}{5} \) and a y-intercept of 4.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation of a straight line. It is a very common method because it offers immediate insight into two key features of the line: the slope and the y-intercept. The general formula for the slope-intercept form is \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) represents the y-intercept.

Why is this form useful? Because it makes graphing lines quick and easy. In one glance, you can see not only where the line crosses the y-axis but also how steep the line is. This simplicity is why the slope-intercept form is often the first method taught in algebra classes.

To use this form, you would follow these steps:

• Identify the slope \( m \).
• Identify the y-intercept \( b \).
• Substitute these values into the equation \( y = mx + b \).

Slope

The slope of a line is a measure of its steepness and direction. It is often referred to as the "rise over run." To imagine this, think of the slope as how much the line rises vertically for every horizontal movement to the right.

Slope is represented by the letter \( m \) in the slope-intercept form \( y = mx + b \). In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope \( m \) can be calculated by the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope means the line moves upward as you move from left to right, while a negative slope means it moves downward. In our initial problem, we see a slope of \( \frac{2}{5} \), indicating a gentle upward slant as the line progresses across the graph.

Understanding the slope helps in predicting the behavior of the line without needing to graph it. It can also aid in solving other mathematical problems involving rates of change.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). This value tells you the exact height at which the line will intersect the y-axis when \( x = 0 \).

For instance, if \( b = 4 \), the line will cross the y-axis at the point \((0, 4)\). This is a critical part of graphing because it gives you a starting point when sketching the line. Once the y-intercept is plotted, you can use the slope to find other points along the line.

In algebra, knowing the y-intercept is handy for both graphing and solving equations, as it provides a simple visual representation of where the line begins its journey across the plane. By understanding the y-intercept and how it connects with the slope, you gain a complete picture of the line's characteristics.