Problem 28 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=-2 x+8 $$

Step-by-Step Solution

Answer

Answer: The slope of the given line is -2, and the y-intercept is 8.

Step 1: Identify the slope

In the given equation, y = -2x + 8, the coefficient of x is -2. This value represents the slope (m) of the line. So, the slope is -2.

Step 2: Identify the y-intercept

In the given equation, y = -2x + 8, the constant term is 8. This value represents the y-intercept (b) of the line. So, the y-intercept is 8.

Step 3: Write the final answer

The slope of the given line is -2, and the y-intercept is 8.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when plotted on a graph. In their most common form, a linear equation is expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

This simple formula tells us that the relationship between the variables is constant, as indicated by the line's slope. In the equation \( y = -2x + 8 \), it fits this standard form of a linear equation, making it easy to identify both the slope and y-intercept by directly analyzing the coefficients and constant term.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is symbolized by \( m \), and it defines how much \( y \) changes for a unit change in \( x \). A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. A zero slope indicates a horizontal line, and an undefined slope is associated with a vertical line.

In the equation \( y = -2x + 8 \), the slope \( m \) is \(-2\). This negative value suggests the line falls as you move along the x-axis. The larger the absolute value of the slope, the steeper the line. Here, a slope of \(-2\) means that for each step you move rightward along the x-axis, the line drops by 2 units in the y-axis direction.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \). In the general form of a linear equation \( y = mx + b \), the letter \( b \) stands for the y-intercept.

In the equation \( y = -2x + 8 \), the y-intercept is \( 8 \). This means that when \( x \) is zero, \( y \) is \( 8 \). On the graph, this is the point (0, 8). The y-intercept plays a crucial role in determining where the line starts on the graph. Understanding this helps us quickly draw a line and comprehend its initial position relative to the y-axis.