Problem 82 - Advanced Engineering Mathematics

Question

In Problems, graph the given equation. $$ x-y-1=0 $$

Step-by-Step Solution

Answer

The graph is a straight line passing through points (0, -1) and (1, 0) with slope 1.

Step 1: Rearrange the Equation

Start by rearranging the given equation in slope-intercept form. The original equation is \(x - y - 1 = 0\). We aim to express it as \(y = mx + b\). Add \(y\) and subtract 1 from both sides to get: \[x - 1 = y\]Therefore, the equation in slope-intercept form is \(y = x - 1\). This shows that the line has a slope \(m\) of 1 and a y-intercept \(b\) of -1.

Step 2: Identify Key Points from the Equation

Using the slope-intercept form \(y = x - 1\), identify key points for graphing. Since the y-intercept \(b = -1\), the point \((0, -1)\) is on the graph. The slope \(m = 1\) means we can move up 1 unit and right 1 unit to find another point. Starting from \((0, -1)\), moving to \((1, 0)\) gives the next point on the line.

Step 3: Plot the Graph

Plot the points \((0, -1)\) and \((1, 0)\) on the Cartesian plane. Use these points to draw a straight line. The line will extend infinitely in both directions, maintaining the same slope (rise over run) of 1.

Step 4: Verify the Graph

Finally, verify by checking if other points satisfy the equation \(y = x - 1\). For instance, using \(x = 2\), compute \(y = 2 - 1 = 1\), so the point \((2, 1)\) satisfies the equation and lies on the line. Ensuring all such points align on the same straight path verifies your graph is accurate.

Key Concepts

Graphing

Graphing linear equations is a fundamental skill in mathematics that allows us to visually interpret relationships between variables. In this exercise, we start by rearranging the equation from its standard form, \(x - y - 1 = 0\), to the more intuitive slope-intercept form. This makes graphing straightforward because it clearly shows the slope and y-intercept.

Once you have the slope-intercept form, \(y = x - 1\), the next step is to identify key points. A good starting point is the y-intercept. From there, use the slope to determine another point on the line. The slope, which is rise over run, tells you how to move from one point to another. In our example, a slope of 1 indicates moving up one unit vertically for every unit you move to the right horizontally. By plotting these points and connecting them with a straight line, you devise a graph that represents the equation.

Verifying your graph by checking additional points ensures its accuracy. You can choose different values for \(x\) and calculate the corresponding \(y\)-values to see if they fit the line, which helps confirm the correctness of your graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\). This format is essential as it clearly showcases two critical elements of a linear graph: the slope \(m\) and the y-intercept \(b\). These elements make graphing an equation much easier.

Let's break down these components:

• The **slope** \((m)\): It shows the steepness of the line. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
• The **y-intercept** \((b)\): This is where the line crosses the y-axis. Knowing this point allows you to begin plotting your line on the graph.

In our example equation \(y = x - 1\), the slope is 1 and the y-intercept is -1. This means the line crosses the y-axis at \(-1\) and rises one unit for each unit it moves right. Transforming your equation into slope-intercept form makes these relationships clear and helps streamline the graphing process.

Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system that forms the foundation of graphing in algebra. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin point \((0,0)\), dividing the plane into four quadrants.

When graphing a linear equation like \(y = x - 1\) on the Cartesian plane:

• Start at the y-intercept \((0, -1)\) on the y-axis, where the line naturally crosses.
• Use the slope to find another point. Here, from \((0, -1)\), you move to \((1, 0)\) by following the slope, which is \(1\) (because it goes up by one for every unit to the right).

By plotting these points and drawing a line through them, you effectively represent the equation. Because the plane is limitless, the line extends in both directions infinitely, always maintaining its slope. Understanding and working with the Cartesian plane empowers you to visualize mathematical relationships effectively.