Problem 3 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ x-y=3 $$

Step-by-Step Solution

Answer

Graph the line with slope 1 through points (0,-3) and (1,-2).

Step 1: Solve for y

To graph a line, it's often easiest to express the equation in slope-intercept form, where it reads as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by solving the equation \(x-y=3\) for \(y\).\[ x - y = 3 \] Subtract \(x\) from both sides to isolate \(-y\):\[ -y = -x + 3 \] Multiply through by \(-1\) to solve for \(y\):\[ y = x - 3 \]. This is now in the form \(y = mx + b\), with \(m = 1\) and \(b = -3\).

Step 2: Identify the Slope and Y-intercept

From the equation \(y = x - 3\), identify the slope \(m\) and the y-intercept \(b\). The slope \(m = 1\), which means the line rises by 1 unit for every 1 unit it moves to the right. The y-intercept \(b = -3\) indicates that the line crosses the y-axis at \(y = -3\).

Step 3: Plot the Y-intercept

Start by plotting the y-intercept on the graph. Locate \(y = -3\) on the y-axis and place a point there. This represents the point \((0, -3)\) on the Cartesian plane, where the line crosses the y-axis.

Step 4: Use the Slope to Find Another Point

Using the slope \(m = 1\), find another point on the line. Since the slope is 1, from the y-intercept \((0, -3)\), move up 1 unit and 1 unit to the right to reach \((1, -2)\). Plot this point on the graph.

Step 5: Draw the Line

Using a ruler, draw a straight line through the two points \((0, -3)\) and \((1, -2)\). Extend the line across the graph in both directions, ensuring it intersects the y-axis where it should.

Key Concepts

Graphing Linear Equations

Graphing linear equations allows us to visually represent relationships between two variables on a coordinate plane. When we graph a linear equation, we typically look for two things: the slope and the y-intercept, which tell us the incline and starting point of the line, respectively.
To graph a line like in the original exercise, you'll usually start with the equation in one of the standard forms, with the slope-intercept form being the most straightforward for graphing. The goal is to find points that the line passes through and then draw a straight line that connects these points.
Here's a simple approach to graphing a line:

• Solve for "y" to get the equation in the slope-intercept form, if needed.
• Identify the y-intercept and plot it on the y-axis.
• Use the slope to find at least one more point on the line.
• Draw a straight line through the points.
  

This process helps you visualize the equation and understand how changes in the equation affect the graph's appearance.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) denotes the y-intercept. This form is incredibly useful for graphing because it gives us both the angle of the line and where it crosses the y-axis directly.
Knowing the slope-intercept form allows you to graph a line quickly by following these easy steps:

• Identify \( m \), the slope, to understand the slant of the line: how much it rises or falls as you move along the x-axis.
• Identify \( b \), the y-intercept, to know where the line crosses the y-axis, which is a key point to plot.

For example, in the exercise's equation \( y = x - 3 \), the slope \( m = 1 \) tells us the line moves up and right at the same rate, and the y-intercept \( b = -3 \) points to its crossing at \( y = -3 \).
This concise form makes it easier to interpret and plot linear equations.

Y-Intercept

The y-intercept is a vital element in understanding linear equations. It is the point where the line crosses the y-axis, represented in the slope-intercept form as \( b \).
Here's how the y-intercept helps you when graphing:

• The value of \( b \) tells you the exact point on the y-axis where the line will pass through, meaning when the x-value is zero, the y-value is \( b \).
• In the example given, the y-intercept is \( -3 \), meaning the point \( (0, -3) \) lies on the graph.

This point is crucial as it is often the starting point in plotting the line. Once you place the y-intercept on the graph, you can use the slope to determine the next points. Understanding this can make graphing easier and faster.

Finding Slope

The concept of slope is central to understanding how a line behaves and how steep it is. The slope is represented by \( m \) in the slope-intercept form \( y = mx + b \) and describes the line's inclination.
To determine the slope:

• Analyze how much the line rises vertically for every unit it moves horizontally.
• This is often written as \( \frac{\text{rise}}{\text{run}} \).

From the example equation \( y = x - 3 \), the slope \( m = 1 \) means you rise 1 unit up for every 1 unit you move to the right. This creates a 45-degree angle line when plotted, assuming equal scaling on the axes.
The slope makes it easy to find other points on the line. Starting from the y-intercept, apply the rise-over-run approach to find another point, ensuring accuracy in the graph you create. This understanding is key to effectively plotting and interpreting linear graphs.