Problem 58 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=4, \quad y\) -intercept \(=5\)

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is \(y=4x+5\).

Step 1: Identify the slope and y-intercept

From the exercise, the slope (m) is 4 and the y-intercept (b) is 5.

Step 2: Write the equation in slope-intercept form

Now, we can plug in the given slope and y-intercept into the slope-intercept form equation. This gives us the equation of the line: \[y = 4x + 5\] So, the equation of the line in slope-intercept form is \(y=4x+5\).

Key Concepts

Line Equations

Line equations are mathematical expressions that describe all the points along a straight line in a coordinate plane. One of the most common forms is the slope-intercept form. The general expression for this type is \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This formula is incredibly useful because it allows us to capture the essence of a line using just two numbers: the steepness of the line (slope) and where it crosses the y-axis (y-intercept).

When writing line equations, one needs to know these two values, making it straightforward to visualize or plot the line on a graph. Knowing the equation also allows various calculations, like predicting the value of \(y\) for any given \(x\). This makes the slope-intercept form particularly favored in algebra and geometry.

Slope

The slope of a line, often represented as \(m\), measures how steep the line is. In algebra, it's defined as the "rise over run," indicating how much \(y\) increases or decreases for every unit increase in \(x\). A slope can be positive, negative, zero, or undefined, each providing a unique line orientation:

• Positive slope: The line ascends from left to right.
• Negative slope: The line descends from left to right.
• Zero slope: Represents a horizontal line (constant \(y\) value).
• Undefined slope: Denotes a vertical line (undefined \(x\) value).

Understanding the slope is crucial because it not only tells us the direction but also the angle at which the line inclines or declines. In our exercise, the slope is \(4\), meaning for every increase of 1 in \(x\), \(y\) increases by 4.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the equation of a line in slope-intercept form, \(y = mx + b\), the y-intercept is represented as \(b\). This value is quite significant as it provides a starting point for graphing the line.

Knowing the y-intercept allows us to easily draw the line on the graph by plotting the intercept and using the slope to find other points. For example, if the equation is \(y = 4x + 5\), the y-intercept is 5, meaning the line crosses the y-axis at (0, 5). This initial fix point is particularly helpful when mapping out the whole line, especially when combined with the slope.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of solving line equations, algebra provides the methods and techniques to rearrange and solve equations like the slope-intercept form. It helps in identifying the components of the equation that correspond to the geometrical properties of the line, like the slope and y-intercept.

By using algebra, we can approach problems systematically, solving for unknowns or transforming equations into different forms for analysis. This makes it an essential skill not just in math, but in a wide range of quantitative disciplines. Working with the equation \(y = 4x + 5\) becomes straightforward with algebraic principles, allowing one to quickly adjust the form of the equation or find specific values of \(y\) for given \(x\) values.