Question

Find the slope and y-intercept of each line. Graph the line. $$ y=5 $$

Short answer

Slope: 0, Y-intercept: 5. Draw a horizontal line at y = 5.

Step-by-step solution

Identify the given equation

The equation given is in the form of a horizontal line. The equation is: \[ y = 5 \]

Determine the slope

For a horizontal line like \( y = 5 \), the slope is 0 because the line does not rise or fall as it moves from left to right. Hence, the slope \( m = 0 \).

Determine the y-intercept

The line intersects the y-axis at the point where \( y = 5 \) and \( x \) can be any value. Therefore, the y-intercept \( b \) is 5.

Graph the line

To graph the line \( y = 5 \), draw a horizontal line that crosses the y-axis at \( 5 \). This line will be parallel to the x-axis and three units higher than the x-axis.

Key concepts

slope of a horizontal line

The concept of the slope is integral in understanding how lines behave on a graph. Slope measures the steepness of a line. Specifically, it quantifies how much the line rises or falls as it moves horizontally across the graph. When we talk about the slope of a horizontal line, like in the equation \(y = 5\), things get straightforward. Horizontal lines do not rise or fall as they move from left to right. Therefore, the change in the y-values (also known as 'rise') is zero, while the change in the x-values (also referred to as 'run') is any non-zero number. Mathematically, the slope is calculated as: \ m = \frac{rise}{run} \. Because the rise for a horizontal line is 0, the slope will always be zero. Consequently, every horizontal line has a slope of zero, indicating that there is no vertical change as one moves horizontally along the line.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. It reveals the value of y when x is zero. For the equation \( y = 5 \), the line crosses the y-axis at y = 5 regardless of the value of x. Thus, the y-intercept here is 5. This means that, as we move along the x-axis, the value of y remains constant at 5. In general, identifying the y-intercept in the equation \(y = c \), where \c\ is a constant, is straightforward. The y-intercept is this constant value c. Calculating the y-intercept can quickly provide a visual clue about where the line will be positioned vertically on the graph.

graphing horizontal lines

Graphing horizontal lines is a simple process once you understand their properties. For the equation \ y = 5\, follow these steps:

• First, locate the y-intercept on the y-axis. In this case, it is 5.
• Next, draw a horizontal line passing through this point. Make sure the line extends in both directions.

This line represents all possible x-values while keeping y constant at 5. It's parallel to the x-axis and does not intersect or curve. Every point on this line has the form \(x, 5\), where x can be any real number. Practicing graphing horizontal lines can instill a visual and intuitive understanding of their constant y-value.

equations of horizontal lines

Identifying and understanding the equations of horizontal lines is a fundamental aspect of algebra. The general form of the equation for a horizontal line is \(y = c\), where \c\ is a constant. This equation signifies that for every value of x along the line, the value of y remains fixed at \c\. Taking \ y = 5\ as an example, no matter what value x takes, y will always be 5. The simplicity of this equation makes it easy to predict and graph the line. When dealing with any horizontal line, remember that:

• The slope is always 0.
• The y-intercept is the constant value in the equation.