Problem 79 - Calculus Volume 1

Question

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. \(4 y+24=0\)

Step-by-Step Solution

Answer

Slope: 0; y-intercept: -6. Graph is a horizontal line at y = -6.

Step 1: Simplify the Equation

First, simplify the given equation to express it in the form of a standard linear equation, which is \( y = mx + b \). The given equation is \( 4y + 24 = 0 \). Solve for \( y \) by isolating it. Subtract 24 from both sides to get \( 4y = -24 \). Then, divide every term by 4 to solve for \( y \). This results in \( y = -6 \).

Step 2: Identify the Slope and the Y-intercept

Now that the equation is in the form \( y = mx + b \), identify the slope \( m \) and the y-intercept \( b \). In the equation \( y = -6 \), there is no \( x \) term, which means \( m = 0 \). The constant term \(-6\) is the y-intercept \( b \). Thus, \( m = 0 \) and \( b = -6 \).

Step 3: Plot the Y-intercept

To graph the line, start by plotting the y-intercept. Since \( b = -6 \), this point is \( (0, -6) \) on the graph.

Step 4: Graph the Horizontal Line

With the slope \( m = 0 \), this line is horizontal. Since it passes through the y-intercept \( (0, -6) \), draw a straight horizontal line across the graph that intersects the y-axis at \( -6 \).

Key Concepts

Slope

When discussing linear equations, the concept of slope is crucial. The slope, often denoted by the letter \( m \), represents the steepness or the incline of a line. It tells you how much the line rises or falls as you move along it horizontally.

• If the slope is positive, the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a perfectly horizontal line; no rise or fall.

In the equation form \( y = mx + b \), the number \( m \) is the slope. For our specific equation \( y = -6 \), there's no \( x \) term, meaning \( m = 0 \). This tells us that the line is horizontal. Whenever you encounter a linear equation without an \( x \) term, remember that you're dealing with a horizontal line.

Y-intercept

The y-intercept is another important feature of linear equations. It is the point where the line crosses the y-axis. This is denoted by \( b \) in the equation \( y = mx + b \).

• The y-intercept provides the starting point if you were to graph the line.
• It is given by the constant term in the equation when \( x \) is zero.

In our equation \( y = -6 \), the y-intercept \( b \) is \(-6\). This means that the line crosses the y-axis at the point \((0, -6)\). Understanding where the line begins on the y-axis helps you quickly and accurately draw the full line on a graph.

Graphing Linear Equations

Graphing linear equations is about creating a visual representation of the equation. This involves plotting points and drawing lines to express the mathematical relationship in the equation. Here are some steps to follow:

• Identify the y-intercept \( b \). Start your graphing at this point.
• Use the slope \( m \) to determine the direction and steepness of the line. Since \( m = 0 \) in our example, this will be a horizontal line.

For the equation \( y = -6 \), you only need the y-intercept because of the zero slope. Plot \((0, -6)\) on the y-axis, then draw a horizontal line passing through this point. This line continues infinitely left and right, always staying level, illustrating the nature of a horizontal slope. By understanding these basics, you'll master graphing any linear equation.