Problem 27 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ x-y=0 $$

Step-by-Step Solution

Answer

The slope is 1, and the y-intercept is (0, 0).

Step 1: Rearrange the Equation

Rearrange the equation from the standard form to the slope-intercept form, which is y = mx + b. The given equation is \( x - y = 0 \). To express it in slope-intercept form, solve for \( y \) by adding \( y \) to both sides: \( y = x \).

Step 2: Identify the Slope and Y-Intercept

In the slope-intercept form \( y = mx + b \), \( m \) is the slope, and \( b \) is the y-intercept. From the equation \( y = x \), identify \( m = 1 \) and \( b = 0 \). Thus, the slope \( m \) is 1, and the y-intercept is \((0, 0)\).

Step 3: Graph the Equation

To graph the equation \( y = x \), start from the y-intercept \((0, 0)\) on the coordinate plane. Since the slope \( m = 1 \), which means for every 1 unit increase in \( x \), \( y \) increases by 1 unit. Draw a straight line through the origin with this slope.

Key Concepts

Understanding Slope

The concept of slope is fundamental in understanding how lines behave on a graph. Slope is a measure of the "steepness" of a line and is represented by the letter \( m \) in the slope-intercept form equation \( y = mx + b \). It tells us how much \( y \) changes for a given change in \( x \). For instance, if the slope \( m = 1 \), this means that for every unit \( x \) increases, \( y \) also increases by one unit. In practical terms,

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope indicates a vertical line.

Understanding slope allows you to quickly determine how a line will look on a graph. It's a crucial tool for any kind of linear graph analysis.

The Role of the Y-Intercept

The y-intercept is the point where the line crosses the y-axis, represented by \( b \) in the slope-intercept form \( y = mx + b \). It occurs when \( x = 0 \), and the y-value at this point is \( b \). In the equation we analyzed, \( y = x \),

• The y-intercept is \( (0, 0) \).
• This means the line passes through the origin of the graph.

The y-intercept is a starting point for graphing a line. Once you have this point, you can use the slope to determine the direction and steepness of the line. Knowing the y-intercept makes it easier to sketch the line accurately on the coordinate plane.

Graphing an Equation

Graphing an equation in slope-intercept form is straightforward if you understand the roles of \( m \) and \( b \). The equation we considered, \( y = x \),

• Has a slope \( m = 1 \) and a y-intercept \( b = 0 \).

To graph this:1. Start at the y-intercept, which is the point \( (0,0) \) here.2. Use the slope to find your next points. Since the slope is 1, move one unit up and one unit right from the y-intercept to find a second point.3. Draw a straight line through these points.When you connect these points with a straight line, you produce a visual representation of the equation. This technique of graphing emphasizes understanding the functional relationship described by the equation, providing insight into how changes in \( x \) affect \( y \). Knowing how to graph accurately ensures you can visualize and interpret linear relationships effectively.