Problem 75 - Calculus Volume 1

Question

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. \(y=2 x-3\)

Step-by-Step Solution

Answer

The slope is 2 and the y-intercept is -3.

Step 1: Identify the Slope-Intercept Form

The given equation is already in the slope-intercept form, which is generally written as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the \( y \)-intercept.

Step 2: Determine the Slope \( m \)

By comparing the given equation \( y = 2x - 3 \) with the slope-intercept form \( y = mx + b \), we identify that the slope \( m \) is \( 2 \).

Step 3: Determine the \( y \)-Intercept \( b \)

In the equation \( y = 2x - 3 \), it is shown that the \( y \)-intercept \( b \) is \( -3 \). This tells us that the line meets the \( y \)-axis at the point (0, -3).

Step 4: Plot the \( y \)-Intercept on the Graph

Start by noting the \( y \)-intercept on the graph. Plot the point (0, -3) on the \( y \)-axis.

Step 5: Use the Slope to Find Another Point

The slope \( m = 2 \) can be rewritten as \( \frac{2}{1} \), indicating a rise of 2 units and a run of 1 unit. From the \( y \)-intercept at (0, -3), move up 2 units and 1 unit to the right to plot another point at (1, -1).

Step 6: Draw the Line

Connect the two points, (0, -3) and (1, -1), with a straight line that extends in both directions. This line represents the graph of the equation \( y = 2x - 3 \).

Key Concepts

Slope-Intercept Form

Linear equations are often expressed in what is known as the "Slope-Intercept Form." This is a straight-line equation that simplifies graphing and analysis. The general formula is \( y = mx + b \), where:

• \( y \) represents the output or dependent variable.
• \( m \) is the slope, showing the rate of change or steepness of the line.
• \( x \) is the input or independent variable.
• \( b \) is the \( y \)-intercept, which is where the graph crosses the \( y \)-axis.

When an equation is in this form, it readily provides both the slope and the \( y \)-intercept, making it easier to graph. This sets the stage for understanding the more detailed aspects of linear equations.

Slope

The slope of a line indicates its steepness and direction. In the equation \( y = 2x - 3 \), the slope \( m \) is 2. Here's what this means:

• For every increase of 1 in \( x \), \( y \) increases by 2. This is known as "rise over run."
• If the slope is positive, like 2, the line rises from left to right.
• A negative slope would mean the line falls from left to right, indicating a decrease as \( x \) increases.
• A slope of 0 results in a horizontal line, where \( y \) stays constant despite changes in \( x \).

The slope is crucial for understanding how changes in one variable affect another. It provides insight into the line's behavior across a graph.

Y-Intercept

The \( y \)-intercept is where the line crosses the \( y \)-axis. In our example equation, \( y = 2x - 3 \), the \( y \)-intercept \( b \) is -3. Here's how to interpret this:

• The line meets the \( y \)-axis at the point \((0, -3)\).
• This point is crucial for graphing as it gives a clear starting point.
• Knowing the \( y \)-intercept allows you to quickly mark the initial point on the graph.
• This helps guide the graphing of the entire line when combined with the slope.

The \( y \)-intercept is a foundational part of graphing linear equations, as it serves as a reference point to apply the slope.

Graphing Linear Equations

Graphing linear equations offers a visual representation of the relationship between variables. To graph \( y = 2x - 3 \):

• Start with the \( y \)-intercept, marked as the point \((0, -3)\).
• Use the slope to find another point. With a slope of 2, move up 2 units and right 1 unit from \((0, -3)\) to reach the point \((1, -1)\).
• Join these two points with a straight line, ensuring it extends in both directions.
• This line represents all the solutions to the equation. Each point on this line is a valid \((x, y)\) pairing that satisfies \( y = 2x - 3 \).

Graphing provides a tangible way to interpret equations, showcasing the line's slope and \( y \)-intercept, making it easier to see changes and predict values.