Problem 21 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The slope of the given line is 9 and the y-intercept is 0.

Step 1: Arranging in Slope-Intercept Form

First, the equation should be rearranged into slope-intercept form \(y = mx + b\). The given equation is \(y-9x=0\). By rearranging this equation by adding 9x to both sides, you get the equation in the slope-intercept form: \(y = 9x + 0\)

Step 2: Identifying the slope

The slope \(m\) of a linear equation in the form \(y = mx + b\) is the coefficient in front of the \(x\)-term. In this equation \(y = 9x + 0\), the slope \(m\) is 9.

Step 3: Identifying the y-intercept

The y-intercept \(b\) of a linear equation in the form \(y = mx + b\) is the constant term. In this equation \(y = 9x + 0\), the y-intercept \(b\) is 0.

Key Concepts

Finding the Slope

The slope of a line is a measure of its steepness, which is defined as the ratio of the rise over the run between two points on the line. In mathematical terms, if you pick two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the context of the equation \( y - 9x = 0 \), once we rearrange it to \( y = 9x \), we identify that the slope \( m \) is 9. This tells us that for every 1 unit increase in \( x \) along the horizontal axis, \( y \) increases by 9 units along the vertical axis, indicating a relatively steep line.

Understanding the Y-intercept

The y-intercept is the point where a line crosses the y-axis. This happens when the \( x \) value is 0. At the y-intercept, the coordinate is always in the form \( (0, b) \), where \( b \) is the y-value. It's the starting point of the line when plotting on a graph.

For the equation given in slope-intercept form \( y = 9x \), the y-intercept is 0. This is seen in the equation as the constant term. Therefore, the y-intercept of this line is the origin \( (0, 0) \) on the graph. This indicates that when \( x \) is 0, \( y \) is also 0, and this is where the line will start on the graph.

Linear Equations and Their Forms

Linear equations represent straight lines on a graph and are characterized by an equation of the first degree. This means that the highest power of the variable is one. There are several forms of linear equations, but one of the most commonly used is the slope-intercept form, \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept.

In cases where equations are not presented in this form, like \( y - 9x = 0 \), they must be rearranged to identify the slope and y-intercept easily. Rearranging involves solving for \( y \) and ensuring it is isolated on one side of the equation. This form is particularly helpful for quickly determining the key characteristics of the line without needing to graph it.

Graphing Linear Equations

Graphing a linear equation involves plotting its line on a coordinate plane. The first step is to find the intercepts or convenient points, then connect these points with a straight line. For our equation \( y = 9x \), the graphing process starts by plotting the y-intercept, which is the point \( (0, 0) \) in this case.

Next, since the slope (rise over run) is 9, you can choose another point on the line by moving 1 unit to the right (along x) and 9 units up (along y), giving the point \( (1, 9) \). Draw a straight line through these points, and you've graphed the equation. Remember to always label your axes and use a ruler for accuracy to make the graph as precise and clear as possible.