Problem 13 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ x-y=0 $$

Step-by-Step Solution

Answer

The graph of the equation is a straight line through points (0,0) and (1,1), with a slope of 1.

Step 1: Rewrite the Equation

Start by rewriting the given equation in slope-intercept form, which is \( y = mx + b \). The original equation is \( x - y = 0 \). Solving for \( y \), we re-arrange the equation: \( y = x \).

Step 2: Identify the Slope and Y-Intercept

In the equation \( y = x \), notice that this can be rewritten as \( y = 1x + 0 \). The slope \( m \) is 1, and the y-intercept \( b \) is 0.

Step 3: Plot the Y-Intercept

On the graph, plot the y-intercept, which is the point where the line crosses the y-axis. For the equation \( y = x \), the y-intercept is at (0, 0).

Step 4: Use the Slope to Plot Another Point

Use the slope of 1 to plot another point. From the y-intercept (0,0), move up 1 unit and to the right 1 unit to the point (1, 1).

Step 5: Draw the Line

Draw a straight line through the points (0, 0) and (1, 1). This line represents the graph of the equation \( y = x \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is an important way of writing the equation of a straight line. It's expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. This form is popular because it clearly defines the line's incline and the point where it crosses the y-axis. To convert any standard linear equation to this form, simply solve for \( y \) to make it the subject. For example, if you have the equation \( x - y = 0 \), rearranging it gives you \( y = x \). Once converted, the slope-intercept form can easily show the rate of change and starting point of a line. This makes it a highly user-friendly form for graphing linear equations.

Slope

The slope of a line in the slope-intercept equation \( y = mx + b \) is denoted by \( m \). It's a measure of how steep the line is and indicates how much \( y \) changes for a change in \( x \). In layman's terms, the slope tells you how quickly a line ascends or descends. There are a few types of slopes to understand:

• A positive slope means the line rises as it moves to the right.
• A negative slope means the line falls as it moves to the right.
• A zero slope indicates a perfectly horizontal line.
• An undefined slope suggests a vertical line.

In our equation \( y = x \), the slope is 1, indicating a gentle rise. This means for every 1 unit you move to the right, the line rises by 1 unit. Understanding slopes helps in determining line behavior from its equation.

Y-Intercept

The y-intercept, found as \( b \) in the equation \( y = mx + b \), is where the line crosses the y-axis. It represents the value of \( y \) when \( x \) is zero. The role of the y-intercept is to set the starting point of the line on a graph. In the equation \( y = x \), the y-intercept is 0, signifying that the line crosses the origin at the point (0,0). Plotting the y-intercept is crucial because it defines one of the points needed to draw the line accurately on a Cartesian plane. Once the y-intercept is located, you can use the slope to find additional points, ensuring a correctly drawn line. Recognizing the y-intercept helps anchor the line visibly on a graph, thereby clarifying its trajectory and origin.