Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(2 y-7=0\)

Step-by-Step Solution

Answer

The slope of the line described by the equation \(2y - 7=0\) is 0, and the y-intercept is 3.5. The line is a horizontal line crossing the y-axis at the point (0, 3.5).

Step 1: Identify the Slope

In the equation \(2y -7 = 0\), you notice there is no \(x\) term. This indicates the slope is 0, because for any value of \(x\), \(y\) remains constant.

Step 2: Solve for y to Find the y-Intercept

Solve the equation for \(y\). Start by adding 7 to both sides of the equation to get \(2y = 7\). Then divide both sides by 2 to solve for \(y\), giving you \(y = 7/2 = 3.5\). Thus, the y-intercept is 3.5.

Step 3: Sketch the Line

The line described by this equation is horizontal, since the slope is 0. The line intersects the y-axis at the point (0, 3.5). To sketch the line, you first draw a horizontal line at \(y = 3.5\) on the coordinate plane. Then mark the y-intercept (0, 3.5).

Key Concepts

Linear Equations

Linear equations form the foundation for understanding how to graph lines on a coordinate plane. A linear equation is an algebraic expression that represents a straight line when graphed. This kind of equation is typically expressed in the form of (y = mx + b), where (m) represents the slope and (b) represents the y-intercept. The slope tells us how the line inclines or declines as we move from left to right on the graph, while the y-intercept signifies the exact point where the line crosses the y-axis.

Understanding the significance of these two components is crucial because they allow us to graph the line quickly and efficiently. For example, if a student is presented with the equation (2y - 7 = 0), by arranging it into the form (y = mx + b), they can instantly identify the slope as 0 and the y-intercept as (7/2) or (3.5). With this information, plotting the line becomes an accessible task.

Graphing Lines

When it comes to graphing lines, understanding the role of the slope and y-intercept is vital. Once you've determined these two values, you can easily plot the line on a coordinate plane. To graph a line with a known slope and y-intercept, start by plotting the y-intercept on the y-axis. From this point, use the slope to determine the direction and steepness of the line.

If the slope is positive, the line will ascend from left to right; if negative, it will descend. If the slope is zero, as it is for the equation (2y - 7 = 0), the line will be perfectly horizontal. After determining the direction, draw a straight line through your points, ensuring it extends equally in both directions for a balanced graph.

Horizontal Line Equation

The horizontal line equation is a special type of linear equation where the slope ((m)) is equal to zero. This happens because the line is parallel to the x-axis and there is no vertical change no matter how much we move along the x-axis. Therefore, the general form of a horizontal line is (y = k), where (k) is any real number representing the constant y-value across the graph.

In our example equation (2y - 7 = 0), solving for (y) gives us (y = 3.5), identifying it as a horizontal line at (y = 3.5) on the graph. This is important to remember: a horizontal line has zero slope and does not rely on (x) for any changes in value.

Y-intercept Calculation

Calculating the y-intercept is an essential step in graphing linear equations. The y-intercept is the point where the line crosses the y-axis, and naturally, its x-coordinate is zero.
For the equation (2y - 7 = 0), the calculation of the y-intercept is straightforward. To find it, isolate (y) by adding 7 to both sides of the equation resulting in (2y = 7), then divide by 2 for (y) to be alone on one side of the equation. The result, (y = 3.5), shows that the y-intercept is at the point (0, 3.5).

Students should keep in mind that calculating the y-intercept requires setting (x) to zero and solving for (y), and this method can be applied to any linear equation to find its y-intercept.