Problem 83 - College Algebra

Question

Describe how to write the equation of a line if two points along the line are known.

Step-by-Step Solution

Answer

The equation of the line is found by first calculating the slope 'm' = (y2 - y1) / (x2 - x1), next calculating the y-intercept 'c' = y1 - m*x1, and then plugging 'm' and 'c' into the equation y = mx + c.

Step 1: Calculate the slope

The slope of a line when two points (\(x_1,y_1\)) and (\(x_2, y_2\)) are known, can be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). This will give the tilt of the line.

Step 2: Find the y-intercept

After we have the slope, we can use the values and the slope obtained to get the y-intercept 'c'. For that, substitute the values of any point (let's say \((x_1,y_1)\)) and m in the formula \(y = mx + c\). It will result in \(c = y_1 - m*x_1\). This gives us the exact point where the line crosses the y-axis.

Step 3: Write the equation of the line

Now that we have both the slope (m) and y-intercept (c), we can represent the equation of the line as \(y = mx + c\).

Key Concepts

Slope Calculation

The slope of a line is a measure of how steep the line is. When two points on the line are given, such as \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) can be calculated using the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula helps us find the ratio of the vertical change to the horizontal change between the two points.
Think of it as the rise over the run. The slope will tell us if the line is increasing, decreasing, or horizontal. If the slope is positive, the line rises to the right. If negative, it falls to the right. If zero, it means the line is perfectly horizontal.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis.
When we find the slope of a line, the next step is to determine this y-intercept.

• Using the equation \( y = mx + c \), we solve for \( c \) by substituting the slope \( m \) and the coordinates of one of the known points, say \( (x_1, y_1) \).
• This gives us:\[ c = y_1 - m \times x_1 \]

With the y-intercept determined, you know exactly where the line meets the y-axis.
It's an important parameter, as it helps in writing the complete equation of the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a method of describing the position of points on a plane using a set of coordinates. Two-dimensional coordinate systems generally involve the x-axis (horizontal line) and y-axis (vertical line).
In this form of geometry, the position of points is determined by two numbers: their x-coordinate and y-coordinate.Understanding coordinate geometry is essential for expressing geometric shapes algebraically.
In the context of our problem, it allows us to express the concept of a line, which is a collection of points, using an equation like \( y = mx + c \).
This equation encompasses both the slope and y-intercept to define every point along that line.

Two-Point Form

The two-point form is a method for finding the equation of a line when two distinct points on the line are known.
These points are usually represented as \( (x_1, y_1) \) and \( (x_2, y_2) \).

• The two-point form equation is derived by first calculating the slope \( m \), and then using one of the known points to find the y-intercept.
• Alternatively, the two-point form of a line can be written directly as:\[ y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1) \]

This representation is particularly helpful when you have limited information, namely the two points, yet need to define the entire line.