Problem 32 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(4,\) passes through the origin

Step-by-Step Solution

Answer

The equation is \( y = 4x \).

Step 1: Understand the Slope-Intercept Form

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

Step 2: Identify Given Values

From the problem, the slope of the line \( m \) is \( 4 \), and the line passes through the origin, meaning the point \( (0,0) \) is on the line.

Step 3: Determine the Y-Intercept

Since the line passes through the origin \((0,0)\), the y-intercept \( b \) is \( 0 \).

Step 4: Write the Equation

Using the slope \( m = 4 \) and the y-intercept \( b = 0 \) in the slope-intercept form formula, the equation of the line is \( y = 4x + 0 \).

Step 5: Simplify the Equation

The equation \( y = 4x + 0 \) simplifies to \( y = 4x \).

Key Concepts

Understanding the Slope-Intercept Form

The slope-intercept form is a fundamental way of expressing a linear equation. It's given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) stands for the y-intercept. This form is particularly useful because it allows you to easily see the slope and the y-intercept right from the equation.
When you write an equation in this form, you can quickly tell how steep a line is and where it crosses the y-axis. This makes graphing the equation straightforward since you can start plotting from the y-intercept and use the slope to determine how the line rises or falls across the graph.

Exploring the Slope

The slope, noted as \( m \), indicates the steepness and direction of a line on a graph. It describes how much \( y \) changes for a given change in \( x \). A positive slope means the line rises as it moves from left to right, while a negative slope indicates the line falls. The slope for a horizontal line is zero, signifying no change in \( y \) as \( x \) changes.
For example, in our linear equation \( y = 4x \), the slope is \( 4 \). This tells us that for every single unit increase in \( x \), \( y \) increases by four units. Knowing the slope is crucial for graphing, as it helps determine the angle and direction of the line.

The Role of the Y-Intercept

The y-intercept, represented by \( b \), is the point where the line crosses the y-axis. In other words, it's the value of \( y \) when \( x \) is zero. The y-intercept is critical in understanding the initial condition or starting point of a line in a graph.
In the equation \( y = 4x \), the y-intercept is \( 0 \). This means that when \( x = 0 \), \( y \) is also \( 0 \). Consequently, the line passes through the origin, \((0,0)\). The y-intercept gives a reference point to start plotting the line, especially when combined with the slope to map out the entire line on the graph.