Problem 13 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=5, b=3 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = 5x + 3 \).

Step 1: Understand the Slope-Intercept Form

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

Step 2: Identify Given Values

In the problem, the given slope \( m \) is 5, and the y-intercept \( b \) is 3.

Step 3: Plug Values into the Equation

Substitute the given values \( m = 5 \) and \( b = 3 \) into the slope-intercept form. This gives us \( y = 5x + 3 \).

Step 4: Write the Final Equation

The equation of the line with the given slope and y-intercept is \( y = 5x + 3 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple and popular way to represent linear equations in mathematics. This format is expressed as \( y = mx + b \). Here, each component of the equation plays a specific role in describing the characteristics of the line. When you see an equation in this form, it allows you to quickly identify two critical aspects of the line:

• The slope \( m \), which indicates how steep the line is.
• The y-intercept \( b \), which is the point where the line crosses the y-axis.

Understanding the slope-intercept form is key because it provides a clear picture of the line's behavior and its position on the coordinate plane. Once you become comfortable with this format, you'll find it much easier to create and analyze linear equations.

Slope

The slope is a measure of the steepness or incline of a line. In the slope-intercept form \( y = mx + b \), \( m \) represents the slope. The slope quantifies the rate of change of the line. Specifically, it shows how much the y-value of a line changes for each unit increase in the x-value. If the slope is positive, the line rises as it moves from left to right. Conversely, a negative slope means the line falls as it progresses.

Here are some key points to remember about the slope:

• A larger absolute value of \( m \) means a steeper line.
• A slope of zero indicates a horizontal line.
• Vertical lines have undefined slopes because they would require division by zero in calculations.

Knowing the slope of a line can help you predict how changes in one variable (often the independent variable "x") will affect another (commonly the dependent variable "y"). It's essential for graphing lines and understanding their direction and inclination.

Y-Intercept

The y-intercept is a fundamental component of the slope-intercept form. It is denoted by \( b \) in the equation \( y = mx + b \). Put simply, the y-intercept is the point where the line crosses the y-axis on a graph. You can determine this by finding the y-value when the x-value is zero. This is often expressed as the coordinate \((0, b)\).

Here are a couple of facts about y-intercepts:

• The y-intercept is always a point on the y-axis.
• It provides a starting point for graphing a line. From this point, you can use the slope to determine other points on the line.

Recognizing the y-intercept's role in a linear equation helps you establish the line's initial position. It acts as a stable anchor point from which the behavior of the line can be further traced based on the slope.