Problem 19 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=0, b=-8 $$

Step-by-Step Solution

Answer

The equation of the line is \(y = -8\).

Step 1: Identify the Equation Form

The equation of a line is generally given in the form of the slope-intercept equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Substitute the Slope

Substitute the given slope \(m = 0\) into the equation \(y = mx + b\). This results in \(y = 0x + b\).

Step 3: Substitute the Y-Intercept

Now, substitute the given y-intercept \(b = -8\) into the equation, replacing \(b\) in \(y = 0x + b\). This gives us \(y = 0x - 8\).

Step 4: Simplify the Equation

Since \(0x\) is equal to zero, the equation simplifies to \(y = -8\). Thus, the equation of the line is \(y = -8\).

Key Concepts

Slope-Intercept Form

A line can be represented using the slope-intercept form, which is very useful in algebra for writing the equation of a line. This form is expressed as \( y = mx + b \). In this equation:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form makes it easy to graph a linear equation or write the equation of a line if you know the slope and y-intercept. For example, in the equation \( y = x + 4 \), the slope \( m \) is 1, and the y-intercept \( b \) is 4. Knowing the slope-intercept form helps in quick visualization and understanding of the line's behavior.

Slope of a Line

The slope of a line, commonly denoted as \( m \), measures the steepness or incline of the line. It shows how much the line rises or falls vertically for every unit it moves horizontally.

• If the slope is positive, the line goes upwards as it moves from left to right.
• If the slope is negative, the line goes downwards as it moves from left to right.
• A zero slope indicates a perfectly horizontal line.
• An undefined slope means a vertical line.

The formula to find the slope if two points on the line are known, \((x_1, y_1)\) and \((x_2, y_2)\), is \( m = \frac{y_2-y_1}{x_2-x_1} \). When the slope is zero, as in \( y = -8 \), the line is horizontal and does not rise or fall at all.
This results in a constant value for \( y \), indicating a flat line along the y-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is a crucial feature of linear equations as it specifies where the line starts on the graph.

• It is represented by \( b \) in the slope-intercept equation \( y = mx + b \).
• The coordinates of the y-intercept are \((0, b)\).

Understanding the y-intercept helps in graphing the line quickly, as it gives a fixed starting point on the vertical axis. For instance, in the exercise \( y = -8 \), the y-intercept is \(-8\). This means the line crosses the y-axis at \( y = -8 \) and remains constant along a horizontal path.