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Chapter 1: Problem 52

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

Short Answer

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

Step by step solution

01

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 \( \frac{5}{4} \).
02

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 \( y + \frac{1}{2} = \frac{5}{4}(x + \frac{3}{2}) \).
03

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 \( y + \frac{1}{2} = \frac{5}{4}x + \frac{15}{8} \).
04

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 \( y = \frac{5}{4}x + \frac{11}{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥� its slope and y-intercept.

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