Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation for the line described. Write an equation for a line parallel to \(g(x)=3 x-1\) and passing through the point (4,9) .

Step-by-Step Solution

Answer

The equation is \( y = 3x - 3 \).

Step 1: Understand the characteristics of parallel lines

Parallel lines have the same slope. The given line is expressed as \( g(x) = 3x - 1 \). By comparing it with the slope-intercept form \( y = mx + b \), identify the slope \( m \) of the line as \( 3 \). A parallel line must have the same slope.

Step 2: Use the slope-point form to write the equation

With the slope \( m = 3 \) and a point \((4, 9)\) on the line, use the point-slope form equation: \( y - y_1 = m(x - x_1) \). Substitute \( x_1 = 4 \), \( y_1 = 9 \), and \( m = 3 \) to get: \( y - 9 = 3(x - 4) \).

Step 3: Simplify the equation

Distribute the \( 3 \) on the right side of the equation: \( y - 9 = 3x - 12 \). Then, add \( 9 \) to both sides to solve for \( y \): \( y = 3x - 3 \).

Step 4: Write the final equation

The equation of the line that is parallel to \( g(x) = 3x - 1 \) and passes through the point \( (4, 9) \) is \( y = 3x - 3 \).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form makes it easier to write the equation of a line. The general formula for the slope-intercept form is \( y = mx + b \), where:

• \( m \) represents the slope of the line. This tells us how steep the line is; a higher value indicates a steeper incline.
• \( b \) is the y-intercept. It's the point where the line crosses the y-axis, and essentially shows the value of \( y \) when \( x \) is 0.

For example, if you have a line equation such as \( y = 3x - 1 \), you can immediately see that the slope \( m \) is 3, and it crosses the y-axis at -1. This setup allows quick identification of these characteristics without needing to manipulate the equation. Slope-intercept form is widely used for its simplicity in visualizing line behavior.

Point-Slope Form

The point-slope form is another way to express a linear equation, especially handy when you know a point on the line and the slope. Its formula is expressed as: \( y - y_1 = m(x - x_1) \), where:

• \( m \) stands for the slope of the line.
• \( (x_1, y_1) \) is a specific point on the line.

Let's say we want to write an equation for a line passing through the point \((4, 9)\) with a slope of 3. Using the point-slope formula: Substitute \( m = 3 \), \( x_1 = 4 \), and \( y_1 = 9 \): \( y - 9 = 3(x - 4) \). This form allows an easy setup when these elements are known, forming the basis for converting into other forms like the slope-intercept if needed.

Equation of a Line

The equation of a line can be formulated in several ways, helping to describe its relationship in a plane. You can say the equation is essentially a rule that tells you how to derive the y-coordinate from a given x-coordinate. When forming the equation of a line, key components include:

• Its slope, which shows how the line rises or falls with respect to the x-axis.
• A point through which it passes, uniquely situating it on the coordinate plane.

To determine the equation of a line parallel to another, it's crucial to match the given line's slope. For instance, finding a line parallel to \( y = 3x - 1 \) means ensuring our new line also has a slope of 3. Incorporating the point-slope form or transforming to the slope-intercept form helps you adjust for specific points the line must pass through.

Coordinates of a Point

Coordinates of a point are the essence of locating points on a coordinate plane, and they consist of a pair of numbers, \((x, y)\). The first number, \(x\), denotes the point's horizontal position, whereas the second number, \(y\), indicates its vertical position. In solving our problem, the point \((4, 9)\) was essential in establishing the line's equation. The point tells us:

• The x-value is 4, marking its position horizontally.
• The y-value is 9, marking its position vertically.

Using these coordinates is crucial for applying the point-slope form of a line. They act as a reference so that any line parallel to a given direction can be tethered to a definite path on the graph. Thus, coordinates are indispensable in determining and describing linear pathways.