Problem 33 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ y=-2 x-5 $$

Step-by-Step Solution

Answer

The slope is -2 and the y-intercept is -5.

Step 1: Identify the slope-intercept form

The general form of a linear equation in slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Identify these components from the given equation \( y = -2x - 5 \).

Step 2: Determine the slope

In the equation \( y = -2x - 5 \), the coefficient of \( x \) is the slope. Therefore, the slope \( m = -2 \).

Step 3: Determine the y-intercept

The \( y \)-intercept is the constant term in the equation, which is \( -5 \). Thus, the \( y \)-intercept \( b = -5 \), meaning the line crosses the \( y \)-axis at the point \( (0, -5) \).

Step 4: Plot the y-intercept

On the graph, plot the point where the line intersects the \( y \)-axis at \( (0, -5) \).

Step 5: Use the slope to find another point

The slope \( m = -2 \) tells us that for every 1 unit increase in \( x \), \( y \) decreases by 2 units. From the point \((0, -5)\), move 1 unit right (to \( x = 1 \)) and 2 units down to reach \( (1, -7) \). Plot this point on the graph.

Step 6: Draw the line

Connect the plotted points \( (0, -5) \) and \( (1, -7) \) with a straight line. Extend this line in both directions, making sure it crosses the \( y \)-axis at \( -5 \).

Step 7: Write Conclusion

The slope of the line is \( -2 \) and the \( y \)-intercept is \( -5 \). The graph is a straight line passing through the points \( (0, -5) \) and \( (1, -7) \).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and represent lines in a coordinate plane. They have a simple structure of the form \( y = mx + b \).This format, known as slope-intercept form, helps us quickly identify two important characteristics: the slope and the \( y \)-intercept.Because the equation is called "linear," it guarantees that its graph will be a straight line. Linear equations typically involve variables raised only to the first power, ensuring the graph is smooth and straight.

• Slope \( (m) \): This number represents the steepness or incline of the line.
• \( y \)-intercept \( (b) \): This is the point where the line crosses the \( y \)-axis.

Linear equations are used to model real-world scenarios where relationships between variables are constant and predictable.Becoming comfortable with identifying these characteristics in an equation empowers you to graph lines easily.

Graphing Linear Equations

Graphing a linear equation involves plotting key points and connecting them to form a straight line. The process begins by recognizing the equation in slope-intercept form \( y = mx + b \).The two primary components—slope \( (m) \) and \( y \)-intercept \( (b) \)—guide how you begin plotting on the graph.To start:

• Identify and plot the \( y \)-intercept point: Find the constant \( b \) in the equation, and plot this point on the \( y \)-axis.
• Use the slope to determine the direction and steepness: The slope tells how many units to move up or down (change in \( y \)) for each unit you move right (change in \( x \)).
• From the \( y \)-intercept, apply the slope: For a slope of \( -2 \), move 1 unit right and 2 units down to plot another point.
• Draw the line: Connect these points with a straight line, extending it in both directions.

This method ensures your graph accurately represents the equation, making it easier to visualize and understand the relationship between \( x \) and \( y \).

Finding Slope and Y-intercept

Finding the slope and \( y \)-intercept is crucial for graphing or understanding linear equations. In the slope-intercept formula \( y = mx + b \), each part—\( m \) and \( b \)—serves a specific purpose.

Determining the Slope \( (m) \):

The slope, represented by \( m \), describes how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.To find the slope, look for the coefficient of \( x \) in the equation.In \( y = -2x - 5 \), the slope is \( -2 \), meaning the line descends as \( x \) increases.

Identifying the Y-intercept \( (b) \):

The \( y \)-intercept, indicated by \( b \), is where the line crosses the \( y \)-axis.It is the value of \( y \) when \( x = 0 \).From \( y = -2x - 5 \), the \( y \)-intercept is \( -5 \), situating the point at \( (0, -5) \) on the graph.Recognizing these components not only facilitates graphing but also provides deeper insights into the equation's representation of relationships in datasets or trends.