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Chapter 0: Problem 10

Finding the Slope of a Line In Exercises \(5-10\) , plot the pair of points and find the slope of the line passing through them. $$ \left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right) $$

Short Answer

Expert verified
The slope of the line passing through the given points is \(-\frac{8}{3}\).

Step by step solution

01

Identify the given points

The two points given in the problem are \(\left(\frac{7}{8}, \frac{3}{4}\right)\) and \(\left(\frac{5}{4},-\frac{1}{4}\right)\) respectively. Convert these points into the format \((x_1, y_1)\) and \((x_2, y_2)\). Therefore, \((x_1, y_1) = \left(\frac{7}{8}, \frac{3}{4}\right)\) and \((x_2, y_2) = \left(\frac{5}{4},-\frac{1}{4}\right)\).
02

Substitute into the formula

Substitute the coordinates of the points into the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This gives \(m = \frac{-\frac{1}{4} - \frac{3}{4}}{\frac{5}{4} - \frac{7}{8}}\).
03

Simplify the expression

Simplify the fraction by separately calculating the numerator and the denominator. The result will be \(m = \frac{-1}{\frac{3}{8}}\). To further simplify, divide -1 by \(\frac{3}{8}\) to get \(m = -\frac{8}{3}\).

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Key Concepts

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

Coordinate Geometry
Coordinate geometry, often referred to as analytic geometry, is a branch of mathematics where the position of points on a plane is described using an ordered pair of numbers. Each point on the plane corresponds to a coordinate
  • The first number in the pair is called the x-coordinate and it represents the horizontal position on the plane.
  • The second number is the y-coordinate, indicating the vertical position.
Coordinate geometry allows visualizing algebraic equations by plotting points and drawing lines. This makes understanding algebraic relationships more intuitive. For example, two points are ight( rac{7}{8}, rac{3}{4} ight ight( and ight( rac{5}{4},- rac{1}{4} ight ight(. By plotting these points on a graph, we can vividly see the relationships and how they form a line.
Slope Formula
The slope formula is a key element in coordinate geometry. It helps determine how steep a line is or the angle at which it inclines or declines. The formula is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where
  • \(m\) is the slope
  • \(x_1, y_1\) are the coordinates of the first point
  • \(x_2, y_2\) are the coordinates of the second point
In our example,
  • The numerator \(-\frac{1}{4} - \frac{3}{4}\) represents the change in vertical position and equals \(-1\).

  • The denominator \(\frac{5}{4} - \frac{7}{8}\) accounts for the horizontal change, simplifying to \(\frac{3}{8}\).
Thus, the slope becomes \(m = \frac{-1}{\frac{3}{8}} = -\frac{8}{3}\).The slope gives a precise measure of the line's inclination.
Linear Equations
Linear equations describe straight lines on a coordinate plane. The standard linear equation is \[ y = mx + b \]where:
  • \(y\) is the dependent variable.
  • \(m\) is the slope, determining the line鈥檚 direction and steepness.
  • \(x\) corresponds to the independent variable.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
In real-life applications, linear equations model straightforward relationships between two variables. Taking the slope \(-\frac{8}{3}\) from the points given as an example, we can use this to write the equation of a line once we have a point through which the line passes. Linear equations thus connect the abstract slope concept with practical line descriptions.
Point-Slope Form
The point-slope form is another efficient way to describe the equation of a line. It is particularly useful when you know the slope of a line and one point on it. The formula is:\[ y - y_1 = m(x - x_1) \]where:
  • \(m\) is the slope.
  • \((x_1, y_1)\) are the coordinates of a specific point on the line.
To illustrate, using the point \((\frac{7}{8}, \frac{3}{4})\) and slope \(-\frac{8}{3}\), the equation becomes\[ y - \frac{3}{4} = -\frac{8}{3}(x - \frac{7}{8}) \].This version of the equation is particularly straightforward when a point and the slope are known, providing a quick way to express linear relationships without needing to find the y-intercept first.

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