Problem 48 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$2 x-5 y=0$$

Step-by-Step Solution

Answer

Slope: \( \frac{2}{5} \), y-intercept: 0. The line passes through the origin.

Step 1: Arrange equation in slope-intercept form

The slope-intercept form of a line equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by rearranging the given equation \( 2x - 5y = 0 \) to this form. We do this by solving for \( y \).

Step 2: Solve for y

To solve for \( y \), first move \( 2x \) to the right side of the equation: \(-5y = -2x\). Then, divide each term by \(-5\) to isolate \( y \): \( y = \frac{2}{5}x \).

Step 3: Identify the slope and y-intercept

Now that the equation is in the form \( y = \frac{2}{5}x + 0 \), we can identify the slope \( m \) as \( \frac{2}{5} \) and the y-intercept \( b \) as \( 0 \).

Step 4: Draw the graph

To graph the line, plot the y-intercept on the y-axis at \( 0 \). From the y-intercept, use the slope \( \frac{2}{5} \), which means "rise 2 units and run 5 units" on the grid, to get the second point. Draw a line through these two points to represent the equation of the line.

Key Concepts

Understanding Slope

The concept of slope is central to understanding the behavior of linear equations. The slope, represented by the symbol \( m \), describes how steep a line is. It gives you a way to measure how one variable changes with respect to another. In its simplest terms, the slope is the 'rise' over the 'run'.

• The 'rise' refers to the change in the vertical direction (y-direction).
• The 'run' refers to the change in the horizontal direction (x-direction).

Mathematically, the slope is calculated using the formula: \[m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\]In our example, the slope is \( \frac{2}{5} \), meaning that for every 5 units you move horizontally (to the right), you move 2 units vertically (up). This positive slope indicates that the line rises as it moves from left to right.

Determining the Y-Intercept

The y-intercept is an essential feature of linear equations in slope-intercept form. It is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. The y-intercept is expressed as \( b \) in the equation \( y = mx + b \).To find the y-intercept:

• Look at the constant term in the slope-intercept equation.
• In the equation \( y = \frac{2}{5}x + 0 \), the y-intercept \( b \) is 0.

This tells us that the line passes through the origin point \( (0,0) \). The y-intercept provides a starting point for graphing the line, from which we can apply the slope to find additional points on the line.

Exploration of Linear Equations

Linear equations form straight lines and are foundational in algebra. They have a standard form known as the slope-intercept form, \( y = mx + b \). This format makes it straightforward to graph linear equations and understand their characteristics.Key characteristics of linear equations include:

• The coefficients \( m \) and \( b \) indicate the slope and y-intercept respectively.
• They represent a constant relationship between two variables, \( x \) and \( y \).

For instance, in our equation \( 2x - 5y = 0 \), we rearranged it to \( y = \frac{2}{5}x + 0 \) to easily identify the slope and y-intercept. Understanding this rearrangement helps in easily graphing the equation by plotting the y-intercept and using the slope to determine the direction and angle of the line. Graphing provides a visual insight into how linear equations behave across different values of \( x \).