Problem 26 - Precalculus Mathematics for Calculus

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

Step 1: Understanding the Slope-Intercept Form

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

Step 2: Substitute the Given Slope

Given that the slope \( m = \frac{2}{5} \), substitute this value into the slope-intercept equation, yielding \( y = \frac{2}{5}x + b \).

Step 3: Substitute the Given Y-Intercept

The problem states that the \( y \)-intercept is 4, which means \( b = 4 \). Substitute this value into the equation: \( y = \frac{2}{5}x + 4 \).

Step 4: Formulate the Final Equation

Having substituted both the slope and the y-intercept values, the final equation of the line is \( y = \frac{2}{5}x + 4 \).

Key Concepts

Equation of a Line

The equation of a line, especially in the slope-intercept form, is fundamental in understanding linear relationships. It provides a direct method to express a line using its slope and y-intercept. The slope-intercept form is written as \( y = mx + b \). Here, \( y \) denotes the dependent variable, commonly representing the vertical axis. \( x \) is the independent variable, representing the horizontal axis.

The equation showcases two crucial aspects of a line:

• The slope \( m \), which indicates the steepness and direction of the line.
• The y-intercept \( b \), which tells us where the line crosses the y-axis.

This form makes it straightforward to graph a line or understand its behavior just by looking at \( m \) and \( b \). Whenever you're given these two pieces of information (the slope and y-intercept), you can easily form the line's equation and predict its path across a graph.

Slope

The slope of a line, represented by \( m \) in the slope-intercept formula \( y = mx + b \), provides key insights into the line's behaviour. It defines the direction and steepness of the line across a graph. If the slope is:

• Positive, the line slants upwards as it moves from left to right.
• Negative, the line slopes downwards as it moves from left to right.
• Zero, the line is perfectly horizontal.
• Undefined, the line is vertical.

For instance, in the equation \( y = \frac{2}{5}x + 4 \), the slope is \( \frac{2}{5} \). This means that for every 5 units you move forward in the \( x \) direction, you move 2 units upward in the \( y \) direction.

This ratio remains constant no matter where you are on the line. Understanding slope is essential for predicting the line's trajectory and for calculating rates of change in various contexts.

Y-Intercept

The y-intercept in a linear equation gives valuable information about the starting point of a line on a graph. It is represented by \( b \) in the equation \( y = mx + b \). Specifically, it indicates the exact point where the line crosses the y-axis.

When you have a y-intercept, it tells you where the line will begin when \( x \) is zero. In the line equation \( y = \frac{2}{5}x + 4 \), the y-intercept is 4.
This means that when you set \( x = 0 \), the value of \( y \) will be 4, so the line crosses the y-axis exactly at point \( (0,4) \).
This information is crucial as it provides a fixed starting reference point for graphing the rest of the line using its slope. Knowing the y-intercept can also help determine the initial condition or fixed value in real-world scenarios such as the starting balance in a bank account or the beginning point of a journey.