Problem 33 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$y=-2 x-5$$

Step-by-Step Solution

Answer

Slope: -2, Y-intercept: -5.

Step 1: Identify the Equation Form

The equation given is \( y = -2x - 5 \). This is in the slope-intercept form of a line, which is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

Step 2: Determine the Slope

By comparing the given equation \( y = -2x - 5 \) to the slope-intercept form \( y = mx + b \), we can identify that the slope \( m \) is \(-2\).

Step 3: Determine the Y-Intercept

Using the same comparison, the y-intercept \( b \) in the equation \( y = -2x - 5 \) is \(-5\). This means the line crosses the y-axis at the point \((0, -5)\).

Step 4: Plot the Y-Intercept

On a graph, start by plotting the y-intercept, which is the point \((0, -5)\). This is where the line will cross the y-axis.

Step 5: Use the Slope to Find Another Point

The slope is \(-2\), which can be written as \(-2/1\). This means that for every 1 unit you move to the right (positive direction on the x-axis), you'll move 2 units down (negative direction on the y-axis). From the point \((0, -5)\), move 1 unit to the right to \((1, -5)\), then 2 units down to \((1, -7)\). Plot this point.

Step 6: Draw the Line

Draw a straight line through the points \((0, -5)\) and \((1, -7)\). Extend the line in both directions, making sure it continues through these points.

Key Concepts

Slope of a Line

The slope of a line describes its steepness and direction. In the slope-intercept form of a linear equation, expressed as \( y = mx + b \), the slope is represented by \( m \). This value tells you how much the \( y \)-coordinate changes when you move along the \( x \)-axis by one unit. A positive slope indicates that the line inclines upward to the right, while a negative slope indicates a decline to the right.

For example, in the equation \( y = -2x - 5 \), the slope \( -2 \) tells us:

• The line decreases by 2 units on the \( y \)-axis for every 1 unit increase on the \( x \)-axis.
• This negative slope reflects downward movement as we move from left to right.

Understanding the slope's value and direction helps in predicting how the line behaves across a graph, enabling accurate plotting and visualization.

Y-Intercept

The \( y \)-intercept is a crucial point where the line crosses the \( y \)-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the \( y \)-intercept. This value indicates the point on the graph where \( x = 0 \). Therefore, the \( y \)-intercept will always have coordinates \((0, b)\).

For the equation \( y = -2x - 5 \):

• The \( y \)-intercept is \( -5 \).
• The line crosses the \( y \)-axis at the point \((0, -5)\).

Locating this intercept provides a starting point for graphing the equation. It ensures that the line is anchored correctly along the \( y \)-axis.

Graphing Linear Equations

Graphing linear equations involves three straightforward steps that make it easy to visualize how equations relate to the coordinate plane. Consider the equation \( y = -2x - 5 \).

Firstly, establish the \( y \)-intercept \((-5)\) on the graph, marking the point \((0, -5)\). This is a key reference. Next, use the slope to determine another point on the line. With a slope of \(-2\), move horizontally 1 unit to the right, and vertically 2 units down from \((0, -5)\) to reach \((1, -7)\).

• Start at the \( y \)-intercept \((0, -5)\).
• Apply the slope: from \((0, -5)\), move right 1 unit and down 2 units to \((1, -7)\).

Finally, draw a straight line through the points \((0, -5)\) and \((1, -7)\). Extend this line across the graph. By following this method, you can accurately graph any linear equation, helping visually interpret the relationship between \( x \) and \( y \) values.