Problem 16 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y-9 x=0 $$

Step-by-Step Solution

Answer

The slope of the graph of the equation \(y - 9x = 0\) is 9, and the y-intercept is 0.

Step 1: Re-arrange the equation to slope-intercept form

Start by making \(y\) the subject of the equation. Add \(9x\) to both sides to balance the equation, resulting in \(y = 9x + 0\).

Step 2: Identify the slope

From the equation \(y = 9x + 0\), the coefficient of \(x\) is the slope \(m\). Thus, the slope \(m\) is 9.

Step 3: Identify the y-intercept

From the equation \(y = 9x + 0\), the constant term present after the \(x\) term is the y-intercept \(b\). Thus, the y-intercept \(b\) is 0.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line. They are fundamental in algebra and can be represented in various forms. The most common is the slope-intercept form:
\[ y = mx + b \]
where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope, and \( b \) is the y-intercept. Linear equations are essential because they model real-world relationships where there is a constant rate of change.
Here's how to identify the key components of a linear equation:

• Look for two variables, typically \( x \) and \( y \).
• The equation expression forms a straight line when graphed.
• The rate of change between \( x \) and \( y \) is constant.

Understanding linear equations is crucial, as they are often used to predict trends, make decisions, and solve problems in various fields like science and business.

Slope

The slope of a line in a linear equation is a number that indicates the steepness and direction of the line. It is represented by the letter \( m \) in the slope-intercept form. The slope is calculated as the "rise over run" or the ratio of the change in \( y \) to the change in \( x \).
Mathematically, this is expressed as:
\[ m = \frac{\Delta y}{\Delta x} \]
In the equation \( y = 9x + 0 \), the slope \( m \) is 9, which tells us:

• The line rises 9 units vertically for every 1 unit it moves horizontally.
• A positive slope means the line ascends as you move from left to right.
• A negative slope would indicate a descending line.

Understanding the slope helps in analyzing how variables relate to each other and is a vital concept in understanding more complex mathematical relationships.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). It tells you the value of \( y \) when \( x \) is zero.
In our example, the equation \( y = 9x + 0 \) gives a y-intercept \( b \) of 0. This means:

• The line crosses the y-axis at the origin (0, 0).
• It signifies the starting value of \( y \) when no changes in \( x \) occur.

Understanding the y-intercept is crucial as it provides insight into the initial condition of a scenario modeled by a linear equation. The y-intercept is often used in analyses to predict initial states or outcomes.