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Chapter 1: Problem 17

Plot the points and find the slope of the line passing through the pair of points. $$ \left(\frac{2}{3}, \frac{5}{2}\right),\left(\frac{1}{4},-\frac{5}{6}\right) $$

Short Answer

Expert verified
The slope of the line passing through the points \(\left(\frac{2}{3}, \frac{5}{2}\right)\) and \(\left(\frac{1}{4},-\frac{5}{6}\right)\) is \(m = -\frac{27}{14}\).

Step by step solution

01

Identify the Given Points

The given points are \(\left(\frac{2}{3}, \frac{5}{2}\right)\) and \(\left(\frac{1}{4},-\frac{5}{6}\right)\). These points will be referred to as Point 1 and Point 2 respectively.
02

Apply the Slope Formula

The formula to calculate the slope (\(m\)) of the line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points into the formula, it becomes \(m = \frac{-\frac{5}{6} - \frac{5}{2}}{\frac{1}{4} - \frac{2}{3}}\).
03

Calculate the Slope

Perform the necessary calculations to determine the slope. Remember that when performing subtraction with fractions, you may need to find a common denominator. After calculating, the slope of this line is \(m = -\frac{27}{14}\).

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Key Concepts

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

Plotting Points on a Coordinate Plane
Plotting points is the first step in visualizing a line on a graph. The coordinate plane is set up with two axes:
  • The horizontal axis, called the x-axis.
  • The vertical axis, called the y-axis.
To plot a point like \( \left( \frac{2}{3}, \frac{5}{2} \right) \), start by finding \( \frac{2}{3} \) on the x-axis. Then, move vertically to \( \frac{5}{2} \) on the y-axis. Repeat this for the second point \( \left( \frac{1}{4}, -\frac{5}{6} \right) \). Mark these points clearly on the graph. Joining these points will represent the line whose slope you need to find.
Understanding how to place and connect these points is crucial to visual interpretation in coordinate geometry.
Understanding the Slope Formula
The slope formula is a core concept in coordinate geometry, used to determine the steepness or incline of a line. The formula is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line. The numerator \(y_2 - y_1\) represents the vertical change, while the denominator \(x_2 - x_1\) signifies the horizontal change. A positive slope means the line ascends from left to right, and a negative slope means it descends. Using the slope formula helps you understand how different points relate in a linear fashion on a graph.
Mastering Fraction Operations
Fraction operations are vital for calculating slopes accurately, especially when dealing with fractional coordinates. When subtracting fractions like \(-\frac{5}{6} - \frac{5}{2}\) or \(\frac{1}{4} - \frac{2}{3}\), it's essential to find a common denominator:
  • To subtract \(-\frac{5}{6} - \frac{5}{2}\), convert to a common denominator of 6: \(-\frac{5}{6} - \frac{15}{6}\).
  • For \(\frac{1}{4} - \frac{2}{3}\), use a common denominator of 12: \(\frac{3}{12} - \frac{8}{12}\).
Performing these operations results in a slope of \(-\frac{27}{14}\). Practicing these skills ensures you're comfortable handling any fractional values in algebraic contexts.
Exploring Coordinate Geometry
Coordinate geometry is the study of geometric figures graphically represented on the coordinate plane. It allows you to understand the spatial relationships between points, lines, and shapes using algebraic formulas. By plotting points and lines, you can solve for characteristics like slopes, distances, and midpoints. This exercise challenges you to delve into these concepts, calculating the slope and representing the relation between two points graphically. Understanding coordinate geometry creates a bridge between visual mathematics and algebraic manipulation, enhancing both problem-solving and interpretation skills.

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Most popular questions from this chapter

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