Problem 3 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(1 ;\) the \(y\) -intercept is \(0 .\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form with a slope of 1 and a y-intercept of 0 is \(y = x\).

Step 1: Identify the Given Values

The equation of a line in slope-intercept form is represented as \(y = mx + b\), where \(m\) stands for the slope and \(b\) for the y-intercept. From the exercise, it's known that the slope \(m\) equals 1 and the y-intercept \(b\) equals 0.

Step 2: Substitute the Given Values into the Equation

Substitute the given values into the slope-intercept form. Replace \(m\) with 1 and \(b\) with 0, yielding the equation \(y = 1x + 0\).

Step 3: Simplify the Equation

The equation \(y = 1x + 0\) can be further simplified. Since multiplying any number by 1 leaves it unchanged, 1x can be written simply as x. Also, adding 0 to anything leaves it unchanged, so +0 can be dropped, leading to the final equation \(y = x\).

Key Concepts

Understanding Linear Equations

In mathematics, a linear equation is one of the most fundamental concepts specifically in the field of algebra. It represents a straight line when plotted on a graph. The general form of a linear equation is \( y = mx + b \) , where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept.

The slope \( m \) indicates how steep the line is, while the y-intercept \( b \) tells us where the line crosses the y-axis. What makes linear equations so essential is their simplicity and the wealth of real-world applications. From calculating finances to predicting trends, the linear model is a first step before advancing to more complex functions.

In the context of the given exercise, we start with the slope-intercept form and insert the known values to draft the equation representing the line. This form is incredibly useful for graphing and analyzing lines quickly.

Deciphering Algebraic Slope

The algebraic slope \( m \) in a linear equation \( y = mx + b \) is a crucial concept as it quantifies the direction and steepness of a line. Think of it as measuring the rate at which the \( y \) values change as the \( x \) values change. A positive slope means the line rises from left to right, while a negative slope indicates the line falls.

A slope of 1, as seen in our exercise, means that for every single unit you move right on the x-axis, you also move up a single unit on the y-axis, creating a 45-degree angle relative to both axes. This slope results in a line that increases at a consistent rate. Understanding slope is critical not only for graphing but also for interpreting the rate of change in various contexts, such as velocity in physics or cost predictions in economics.

The Significance of the Y-Intercept

In the linear equation \( y = mx + b \) , the y-intercept \( b \) is where the line crosses the y-axis. Simply put, it is the value of \( y \) when \( x \) is zero. This point is fundamental because it gives you a starting point for graphing the line and also offers valuable insight into the context of the problem.

For instance, if the y-intercept is 0, as it is in our exercise, it means that the line passes through the origin of the graph, which is the point \( (0, 0) \) . A zero y-intercept in real-life situations might indicate that there is no baseline value or initial cost when no quantities are present. The y-intercept helps in quickly visualizing the line and determining its position relative to the y-axis on a coordinate graph.