Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Slope } \pi \text { , passing through }(3,5) $$

Step-by-Step Solution

Answer

The equation of the line is \( y = \pi x + (5 - \pi * 3) \)

Step 1: Identify the slope and the coordinates of the point

The slope 'm' is given as \( \pi \), and the point through which the line passes is (3,5). These values will be used in the standard equation of the line, which is \( y = mx + c \), to find the equation of the line.

Step 2: Substitute 'm' and the coordinates into the line equation to find 'c'

Substitute the slope \( \pi \) for 'm' and the point (3,5) for (x, y) in the equation \( y = mx + c \), the equation becomes: \( 5 = \pi * 3 + c \). By solving this equation, 'c' could be found.

Step 3: Find the value of 'c'

To solve the equation \( 5 = \pi * 3 + c \) for 'c', subtract \( \pi * 3 \) from both sides of the equation. This leads to: \( c = 5 - \pi * 3 \).

Step 4: Substitute 'm' and 'c' into the line equation

Now that we have both 'm' and 'c', substitute them back into the initial line equation \( y = mx + c \). The equation of the line becomes: \( y = \pi x + (5 - \pi * 3) \).

Key Concepts

Slope

The slope of a line is a number that describes how steep or flat the line is. It tells us how much the line rises or falls as we move from one point to another on the line. Slope is often represented as 'm' in mathematical formulas. It is calculated as the change in the vertical direction (rise) divided by the change in the horizontal direction (run).

For any two points

• (x₁, y₁) and (x₂, y₂) on a line, the slope 'm' is calculated as: \( m = \frac{y₂ - y₁}{x₂ - x₁} \)

A positive slope means the line is increasing, a negative slope indicates it's decreasing, and a zero slope implies a flat line. The slope of a vertical line, however, is undefined since there is no horizontal change. In the given problem, the slope is given as \( \pi \), which is an unusual number for a slope, but it works just like any other number when solving for the line's equation.

Linear Equation

A linear equation describes a straight line in algebra. The most common form you'll come across is the slope-intercept form, which is \( y = mx + c \). Here, 'y' and 'x' are variables that represent any point on the line.

• 'm' is the slope, which tells us the direction and steepness of the line.
• 'c' is the y-intercept, which is the point where the line crosses the y-axis.

With a known slope and a point on the line, you can find the entire equation of the line. By substituting these values into the linear formula, you determine the y-intercept 'c'. In our solution, with slope \( \pi \), and using the point (3, 5), the y-intercept is calculated as \( 5 - \pi \times 3 \) to complete the equation \( y = \pi x + (5 - \pi \times 3) \).

Point-Slope Form

The point-slope form of a linear equation is a convenient way to write the equation of a line when you know one point on the line and the slope. The formula is

• \( y - y₁ = m(x - x₁) \)

where \((x₁, y₁)\) is a specific point the line passes through, and 'm' represents the slope.

This form highlights how a line can be defined by a single point and its slope, making it easier to understand and apply in different problems. When given a slope \( \pi \) and a point (3, 5), you could directly apply these into the point-slope form: \( y - 5 = \pi(x - 3) \). This gives an alternative starting point compared to rewriting it in the slope-intercept form, showing flexibility in linear equations.