Problem 2 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=9 x+4$$

Step-by-Step Solution

Answer

The slope of the line is 9 and the y-intercept of the line is 4.

Step 1: Identify the Slope

The slope is the coefficient of \(x\) in the given equation. So, by comparing the equation \(y = 9x + 4\) with the standard form \(y = mx + b\), the slope (\(m\)) is 9.

Step 2: Identify the Y-intercept

The \(y\)-intercept is the constant term in the given equation. So, from the equation \(y = 9x + 4\), the \(y\)-intercept (\(b\)) is 4.

Key Concepts

Slope of a Line

In a linear equation, the slope is crucial as it defines how steep a line is. Slope is often denoted by the letter \( m \). It's specifically expressed as the 'rise over run', meaning how much \( y \) increases or decreases as \( x \) does by one unit.
In simple terms, it's the rate of change between the variables. For the equation \( y = 9x + 4 \), the slope \( m \) is 9. This indicates two things:

• The line goes up steeply since 9 is quite a large number.
• For every one unit increase in \( x \), \( y \) increases by 9 units.

Recognizing the slope can help predict how the line behaves across its path on the graph.

Y-intercept

The \( y \)-intercept is another vital component in the context of linear equations. It represents the precise point where the line crosses the \( y \)-axis of the graph. In the equation \( y = 9x + 4 \), this is the constant term which is 4.
This \( y \)-intercept is easy to spot as it doesn't involve the variable \( x \). Instead, it’s the value of \( y \) when \( x \) is zero. In a practical view, when you pinpoint x at zero, you're essentially locating where the line begins on the vertical axis.
This can be visualized simply as:

• If you imagine walking along the \( y \)-axis, you’d start at \( y = 4 \).
• The line will cross or intercept it at this height.

Understanding the \( y \)-intercept allows us to figure out the start point of the line on a graph.

Standard Form of a Linear Equation

The standard form of a linear equation is a fundamental aspect of algebra. It is typically written as \( Ax + By = C \). While the form \( y = mx + b \), also known as the slope-intercept form, is often preferred for its clear depiction of slope and intercept, the standard form is useful in other scenarios.
Here, \( A \), \( B \), and \( C \) are integers, with \( A \) and \( B \) not both being zero. Let's explore what makes the standard form special:

• It can be beneficial for determining or comparing different lines in a multi-line equation scenario.
• Helps managing equations in problems involving linear combinations or intersection calculations.

Though the given equation \( y = 9x + 4 \) is in slope-intercept form, transforming it to standard form would look like \( -9x + y = 4 \). Knowing both forms enriches our mathematical versatility.