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Chapter 2: Problem 37

Find the slope of the line that passes through the given points, if possible. See Example 2. $$ \left(\frac{1}{4}, \frac{9}{2}\right),\left(-\frac{3}{4}, 0\right) $$

Short Answer

Expert verified
The slope of the line is \( \frac{9}{2} \).

Step by step solution

01

Identify the Coordinates

The points given are \( \left( \frac{1}{4}, \frac{9}{2} \right) \) and \( \left( -\frac{3}{4}, 0 \right) \). These can be labeled as \( (x_1, y_1) \) and \( (x_2, y_2) \). Here, \( x_1 = \frac{1}{4} \), \( y_1 = \frac{9}{2} \), \( x_2 = -\frac{3}{4} \), and \( y_2 = 0 \).
02

Use the Slope Formula

The slope \( m \) of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
03

Substitute the Values into the Formula

Substitute the values from the coordinates into the formula:\[ m = \frac{0 - \frac{9}{2}}{-\frac{3}{4} - \frac{1}{4}} \]
04

Calculate the Numerator

The numerator of the slope formula is \( 0 - \frac{9}{2} = -\frac{9}{2} \).
05

Calculate the Denominator

Calculate the denominator:\( -\frac{3}{4} - \frac{1}{4} = -\frac{3+1}{4} = -\frac{4}{4} = -1 \).
06

Simplify the Slope Calculation

Substitute the calculated numerator and denominator:\[ m = \frac{-\frac{9}{2}}{-1} = \frac{9}{2} \]The negative signs cancel each other.

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Key Concepts

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

Slope Formula
The slope of a line is a measure of its steepness and direction. It's crucial in understanding how two points are related by a linear path. To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the Slope Formula:
  • The formula is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
  • The numerator \(y_2 - y_1\) represents the change in \(y\) values, while the denominator \(x_2 - x_1\) is the change in \(x\) values.
The slope formula essentially compares the vertical change (rise) to the horizontal change (run) between two points. If the numerator and denominator are both negative, their signs cancel each other out, making the slope positive. Conversely, if one is negative and the other positive, the slope is negative, indicating the line falls as you move left to right.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to analyze and solve geometrical problems. Here, we represent points on a two-dimensional plane with pairs of numbers.
  • The first number in the pair is the x-coordinate and indicates horizontal placement.
  • The second number is the y-coordinate and shows vertical placement.
By using ordered pairs like \((x, y)\), we can easily depict points on the Cartesian plane. This system allows for precise location plotting, which helps in various tasks such as finding slopes and equations of lines. Coordinate geometry creates a visual way to handle algebraic expressions, making it easier to understand concepts such as the slope of a line or the shape of a curve.
Linear Equations
Linear equations are equations that graph as straight lines on a coordinate plane. They are fundamental in algebra, showing how variables interact under constant rates of change. The general form is:
  • \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • The slope \(m\) indicates the line's steepness, while the intercept \(b\) is where the line crosses the y-axis.
These equations are versatile, used to model real-life scenarios such as predicting trends, creating budgets, and even physics problems involving constant speeds. When working with linear equations, one can quickly determine relationships between two variables, significantly aiding problem-solving in both simple and complex problems. Understanding the relationship of slopes within these equations is especially valuable in determining parallel and perpendicular lines.

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Most popular questions from this chapter

Graph each equation. \(y=\frac{x-3}{|x-3|}\)

Fire Protection. City growth and the number of fires for a certain city are related by a linear equation. Records show that 113 fires occurred in a year when the local population was \(150,000\) and that the rate of increase in the number of fires was 1 for every \(1,000\) new residents. a. Using the variables \(p\) for population and \(F\) for fires, write an equation (in slope-intercept form) that the fire department can use to predict future fire statistics. b. How many fires can be expected when the population reaches \(200,000 ?\)

Find \(g(2)\) and \(g(3)\). \(g(x)=2 x^{2}-x+1\)

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. $$ f(x)=(x-1)^{3} $$

Complete table. \(g(b)=2\left(-b-\frac{1}{4}\right)\) \(\begin{array}{|r|l|}\hline \boldsymbol{b} & \boldsymbol{g}(\boldsymbol{b}) \\\\\hline-\frac{3}{4} & \\\\\frac{1}{6} & \\\\\frac{5}{2} & \\\\\hline\end{array}\)
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