Problem 52 - Calculus with Concepts in Calculus

Question

Find a two-point equation of the given line. The line containing \(\left(-\frac{3}{2},-\frac{1}{2}\right)\) and \(\left(\frac{1}{2}, 2\right)\)

Step-by-Step Solution

Answer

The two-point equation of the line is \( y = \frac{5}{4}x + \frac{11}{8} \).

Step 1: Calculate Slope (m)

First, find the slope of the line that passes through the points \( \left(-\frac{3}{2}, -\frac{1}{2}\right) \) and \( \left(\frac{1}{2}, 2\right) \). The formula for the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the values: \( m = \frac{2 - \left(-\frac{1}{2}\right)}{\frac{1}{2} - \left(-\frac{3}{2}\right)} = \frac{2 + \frac{1}{2}}{\frac{1}{2} + \frac{3}{2}} = \frac{\frac{5}{2}}{2} = \frac{5}{4} \). The slope of the line is \( \frac{5}{4} \).

Step 2: Use Point-Slope Form

Plug the slope and one of the points, say \( \left(-\frac{3}{2}, -\frac{1}{2}\right) \), into the point-slope form equation: \( y - y_1 = m(x - x_1) \). Substituting, we have \( y + \frac{1}{2} = \frac{5}{4}(x + \frac{3}{2}) \).

Step 3: Simplify the Equation

Distribute the slope on the right to simplify: \( y + \frac{1}{2} = \frac{5}{4}x + \frac{5}{4} \times \frac{3}{2} \). Calculate \( \frac{5}{4} \times \frac{3}{2} = \frac{15}{8} \). So the equation becomes \( y + \frac{1}{2} = \frac{5}{4}x + \frac{15}{8} \).

Step 4: Convert to Two-Point Form

Finally, rewrite the equation in two-point form. Subtract \( \frac{1}{2} \) from both sides to solve for \( y \): \( y = \frac{5}{4}x + \frac{15}{8} - \frac{4}{8} = \frac{5}{4}x + \frac{11}{8} \). So, the equation of the line is \( y = \frac{5}{4}x + \frac{11}{8} \).

Key Concepts

Slope Calculation

To find the two-point equation of a line, the first step is calculating the slope. The slope, represented by \( m \), is an essential component of the line equation. It tells us the "steepness" or the "incline" of the line.
The formula used for slope calculation is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates how much the line travels vertically (rise) for a unit of horizontal travel (run).

• Identify the coordinates of the two points. Here, they are \( \left(-\frac{3}{2}, -\frac{1}{2}\right) \) and \( \left(\frac{1}{2}, 2\right) \).
• Substitute these into the slope formula: \( m = \frac{2 - \left(-\frac{1}{2}\right)}{\frac{1}{2} - \left(-\frac{3}{2}\right)} \).
• Simplify: \( m = \frac{\frac{5}{2}}{2} = \frac{5}{4} \).

The outcome \( \frac{5}{4} \) is the slope, indicating the direction and steepness of the line.

Point-Slope Form

Once the slope is determined, the next step involves using the point-slope form to construct the line's equation. This form is a handy way to draft an equation when you know one point on the line and its slope.
The point-slope form is expressed as \( y - y_1 = m(x - x_1) \).

• Choose one of the points, say \( \left(-\frac{3}{2}, -\frac{1}{2}\right) \).
• Plug the slope \( \frac{5}{4} \) and the chosen point into the formula, giving us:
  \( y + \frac{1}{2} = \frac{5}{4} (x + \frac{3}{2}) \).
• This step establishes the foundation of the line's equation, connecting the calculated slope with the known coordinates of a point on the line.

Equation Simplification

The final goal is to simplify this provisional line equation into a more universally recognized form, typically the slope-intercept form \( y = mx + b \).
Simplification involves expanding and rearranging terms to isolate \( y \).

• Distribute the slope term on the right-hand side:
  \( y + \frac{1}{2} = \frac{5}{4}x + \left(\frac{5}{4} \times \frac{3}{2}\right) \).
• Calculate \( \frac{5}{4} \times \frac{3}{2} = \frac{15}{8} \).
• Then the equation simplifies to \( y + \frac{1}{2} = \frac{5}{4}x + \frac{15}{8} \).
• To solve for \( y \), subtract \( \frac{1}{2} \) from both sides:
  \( y = \frac{5}{4}x + \frac{15}{8} - \frac{4}{8} \).
• Further simplification gives \( y = \frac{5}{4}x + \frac{11}{8} \).

This neat form \( y = \frac{5}{4}x + \frac{11}{8} \) is the simplified equation, ready to be used and interpreted, providing insights into the line's attributes – its slope and y-intercept.