Problem 32 - Prealgebra

Question

The problems below review material we covered in Section 4.9 Graph each equation. $$x-y=3$$

Step-by-Step Solution

Answer

Graph a line with slope 1 and y-intercept -3.

Step 1: Understand the Equation Format

The given equation is a linear equation in two variables: \(x-y=3\). In this form, it's clear we are dealing with a straight line. Rewrite the equation in a more familiar format if necessary.

Step 2: Convert to Slope-Intercept Form

Convert the equation \(x-y=3\) to the slope-intercept form, which is \(y = mx + b\). To do this, solve for \(y\): add \(y\) to both sides to get \(x = y + 3\), then subtract 3 from both sides to get \(y = x - 3\).

Step 3: Determine the Slope and Y-Intercept

From the equation \(y = x - 3\), we see that the slope \(m\) is 1, and the y-intercept \(b\) is -3. This informs us that the line rises 1 unit vertically for every 1 unit it moves horizontally, and it crosses the y-axis at -3.

Step 4: Plot the Y-Intercept

Start by plotting the y-intercept point (0, -3) on the graph. This is where the line crosses the y-axis.

Step 5: Use the Slope to Plot Another Point

From the y-intercept (0, -3), use the slope of 1 to plot another point. Move 1 unit to the right (positive x-direction) and 1 unit up (positive y-direction) to plot the next point (1, -2).

Step 6: Draw the Line

With the points (0, -3) and (1, -2) plotted, draw a straight line through these points to represent the equation \(y = x - 3\). Extend the line across the graph in both directions.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most commonly used ways to express a linear equation. It's written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) indicates the y-intercept. The slope \( m \) shows how steep the line is: it represents the rate of change between the variables \( x \) and \( y \). It's defined as the rise over run, meaning how much \( y \) changes for a change in \( x \).

In our case, the original equation \( x - y = 3 \) was converted to \( y = x - 3 \) to fit this form. Here, the slope \( m \) is 1, signaling that for each unit x increases, y increases by the same amount.

The beauty of the slope-intercept form is its straightforwardness. It allows you to easily identify the slope and the y-intercept by just looking at the equation. This makes it perfect for quickly sketching graphs without much computation.

Y-Intercept

The y-intercept is a crucial concept in graphing linear equations. It is the point where the line crosses the y-axis. To find the y-intercept in the slope-intercept form \( y = mx + b \), you simply look at the value of \( b \). In our example equation \( y = x - 3 \), the y-intercept is -3.

This means the line intersects the y-axis at the point (0, -3). The y-intercept is significant because it provides a starting point for graphing the equation. You begin plotting your line on a graph by marking this intercept, which makes it easier to apply the slope to find other points on the graph. Remember, the x-coordinate is always 0 at the y-intercept, providing a handy anchor when plotting.

Graphing Steps

Graphing linear equations involves a few systematic steps. Once you have the slope-intercept form, it's straightforward.

• Start by plotting the y-intercept on the graph. For the equation \( y = x - 3 \), you would plot the point (0, -3).
• Then, use the slope to determine additional points. The slope of 1 tells you to move 1 unit to the right and 1 unit up from the y-intercept, leading to the point (1, -2).
• After these points are plotted, draw a line through them. Extend this line across the graph.

Each of these steps helps ensure accuracy in your graph. By first identifying and plotting the y-intercept, then using the slope, you efficiently find multiple points on the line without error. This method allows for visually understanding the relationship between \( x \) and \( y \) as described by the linear equation.