Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

\text { Does }(3,9) \text { lie above or below the line } y=3 x-1 ?

Step-by-Step Solution

Answer

The point (3,9) lies above the line.

Step 1: Understand the equation

The line equation given is \( y = 3x - 1 \). This equation represents a straight line where the slope is 3 and the y-intercept is -1.

Step 2: Evaluate y for the x-coordinate

Using the x-coordinate of the point (3,9), we substitute \( x = 3 \) into the line equation to find the corresponding y-value on the line. Thus, \( y = 3(3) - 1 = 9 - 1 = 8 \).

Step 3: Compare the y-coordinate

The y-value we calculated for the point on the line is 8. The y-coordinate of the given point is 9. Since 9 > 8, the point (3,9) lies above the line \( y = 3x - 1 \).

Key Concepts

Understanding Linear Equations

Linear equations form the basis for many concepts in coordinate geometry. They represent relationships between two variables in a graph, typically seen in the xy-plane. A linear equation like \( y = 3x - 1 \) expresses how the value of \( y \) depends on \( x \). The general form of a linear equation is \( y = mx + c \), where:

• \( m \) represents the slope of the line.
• \( c \) is the y-intercept, where the line crosses the y-axis.

Linear equations are called 'linear' because they graph as straight lines. Every point on such a line is a solution to the equation. This means the x and y values of any point (x, y) satisfy the equation.
In problems involving points and lines, you're often tasked to find relationships between points and the line dictated by the equation.

The Slope-Intercept Form

The slope-intercept form is a specific way to write linear equations. In this form, an equation is expressed as \( y = mx + c \). This method of writing is popular because it immediately gives you the slope \( m \) and the y-intercept \( c \).
Understanding the slope and intercept gives a clear picture of the line's behavior:

• The slope \( m \) tells us how steep the line is. A larger slope means a steeper line.
• A positive \( m \) means the line rises as it moves from left to right.
• A negative \( m \) indicates that the line falls from left to right.
• The y-intercept \( c \) is where the line crosses the y-axis, revealing the value of \( y \) when \( x = 0 \).

By interpreting the slope-intercept form, you can quickly determine these characteristics and visualize the line even before plotting it.

Analyzing the Point-Line Relationship

In coordinate geometry, understanding the relationship between a point and a line is crucial. This involves determining where the point stands relative to the line, either above, below, or on it. To evaluate a point-line relationship, follow these steps:

• Calculate the y-value on the line for a given x-coordinate of the point using the line's equation.
• Compare the y-value of the line and the y-coordinate of the point.

If the y-coordinate of the point is greater than the y-value from the line, the point is above the line. If it is less, the point is below the line. If they are equal, the point lies on the line. This comparison tells you about the position of the point in relation to the line, helping in drawing conclusions such as the point’s proximity, alignment, or spatial relation to the line.