Problem 25 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=9 x+1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is \(9\), and the y-intercept is \(1\).

Step 1: Identify the slope

In the given equation, (\(y=9x+1\)), we can see that it is already in the slope-intercept form (\(y=mx+b\)). So, we can easily identify the slope by looking at the coefficient of \(x\). In this case, the slope (\(m\)) is \(9\).

Step 2: Identify the y-intercept

The y-intercept can also be easily identified from the given equation. In the slope-intercept form (\(y=mx+b\)), the y-intercept is the constant term (b). In the given equation (\(y=9x+1\)), the constant term (b) is \(1\). Therefore, the y-intercept is \(1\).

Step 3: Write down the slope and y-intercept

Now that we have identified the slope and y-intercept, we can write them down. The slope (\(m\)) of the line is \(9\), and the y-intercept (\(b\)) is \(1\).

Key Concepts

Understanding the Slope of a Line

When it comes to linear equations, the slope of a line is a measure of its steepness or the angle it makes with the horizontal. Specifically, the slope is the ratio of the rise (the change in the vertical direction) to the run (the change in the horizontal direction). A larger absolute value of the slope indicates a steeper line.

In mathematical terms, if you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \.
When we have a linear equation in slope-intercept form, which looks like \( y = mx + b \), identifying the slope becomes even simpler. The coefficient \( m \) in front of the variable \( x \) is the slope. This form is designed to make it easy to see both the slope and the y-intercept at a glance. In the exercise provided, the equation \( y = 9x + 1 \) clearly shows that the slope is \( 9 \) without needing further calculation.

Deciphering the Y-Intercept

The y-intercept of a graph is where the line crosses the y-axis. This is typically when \( x = 0 \) and can be viewed as the starting point of the line on the graph. The y-intercept is a fundamental part of understanding linear equations because it provides a point through which the line will pass — giving us a clear point to plot on a graph.

In the equation \( y = mx + b \), \( b \) represents the y-intercept. If you set \( x \) to zero in this equation, the resulting value of \( y \) will always be the y-intercept. In our exercise, where the equation is \( y = 9x + 1 \), the y-intercept is identified as \( 1 \) which means the line crosses the y-axis at the point \( (0, 1) \) on the coordinate plane.

Linear Equations and Their Graphs

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. Most commonly, linear equations are presented in two forms: slope-intercept form \( y = mx + b \) and standard form \( Ax + By = C \). The slope-intercept form is especially helpful for graphing since it directly gives you the slope and y-intercept.

The slope tells us how to draw the line after we have marked the y-intercept on our graph. For every unit we move to the right along the x-axis (our run), we move \( m \) units up or down (depending on the sign of the slope) along the y-axis (our rise). For the given equation \( y = 9x + 1 \) from the exercise, we start at the y-intercept \( (0, 1) \) and then, because the slope is \( 9 \) (a positive number), we would go up 9 units for every 1 unit we go to the right to continue plotting the line.