Problem 40 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (2,9) and (8,6)\(;\) use slope-intercept form.

Step-by-Step Solution

Answer

Equation: \( y = -\frac{1}{2}x + 10 \)

Step 1: Find the slope

To find the slope (m) of the line that passes through the points (2, 9) and (8, 6), use the formula for the slope of a line, which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the coordinates (2, 9) as \((x_1, y_1)\) and (8, 6) as \((x_2, y_2)\), we get: \[ m = \frac{6 - 9}{8 - 2} = \frac{-3}{6} = -\frac{1}{2} \] So, the slope of the line is \(-\frac{1}{2}\).

Step 2: Use the slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We have determined that \( m = -\frac{1}{2} \). We now use one of the given points to find \( b \). Utilizing the point (2, 9): \[ 9 = -\frac{1}{2} \times 2 + b \] \[ 9 = -1 + b \] Add 1 to both sides to solve for \( b \): \[ b = 10 \]

Step 3: Write the equation

Substitute \( m = -\frac{1}{2} \) and \( b = 10 \) back into the slope-intercept equation to find the equation of the line: \[ y = -\frac{1}{2}x + 10 \] This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

An equation of a line represents a straight path on a coordinate plane, describing every point that lies on it. There are different forms to express the equation of a line, each serving its purpose and utility in mathematics. Particularly, since our task involves using the slope-intercept form which is

• Slope-Intercept Form: This is represented by the equation: \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

The slope tells us how steep the line is, while the y-intercept is the point where the line crosses the y-axis. These components are essential because they allow us to draw the entire line with just these two pieces of information.
Once you've identified the values for \( m \) and \( b \), you can write a complete equation representing the line.

Mathematical Concepts

In understanding how to arrive at the equation of a line, some key mathematical concepts become useful. First, let's discuss the idea of slope. The slope is a measure of how "tilted" the line is. The formula for slope is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This helps in finding the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
Another concept is using the derived slope in forming the equation of the line. After finding the slope, substitution into the slope-intercept form helps to find the y-intercept, \( b \).
Together, these operations allow us to express behaviors and features of linearity through simple algebraic expressions.

Coordinate Geometry

Coordinate geometry is an area of mathematics where algebra and geometry converge to describe space using coordinates. In the task given, we are using coordinate geometry to find the equation of a line in a plane, specifically by finding the slope through two points and subsequently creating the equation. When working with

• Points: Represented as (x, y) coordinates, each describing a distinct location on the plane.
• Lines: Described through equations, which link algebraic expressions to geometric lines that we can graphically visualize.

The intersection of these defines positional relationships in space. The ability to determine an equation of a line by analyzing the slope between two coordinates is a fundamental application of coordinate geometry, allowing us to graphically depict these algebraic solutions on a coordinate plane.