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Chapter 2: Problem 39

Find the slope of the line that passes through the given points, if possible. See Example 2. $$ (0.7,-0.6),(-0.9,0.2) $$

Short Answer

Expert verified
The slope of the line is \(-0.5\).

Step by step solution

01

Understand the Slope Formula

The slope of a line measures how steep the line is and can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
02

Identify Coordinates

Identify the coordinates of the given points. The first point has coordinates \((x_1, y_1) = (0.7, -0.6)\) and the second point has coordinates \((x_2, y_2) = (-0.9, 0.2)\).
03

Apply Slope Formula

Substitute the coordinates into the slope formula. Use \(x_1 = 0.7\), \(y_1 = -0.6\), \(x_2 = -0.9\), and \(y_2 = 0.2\). The calculation is: \[m = \frac{0.2 - (-0.6)}{-0.9 - 0.7} \]
04

Simplify Numerator

Simplify the numerator of the slope formula. Calculate \(0.2 - (-0.6)\), which becomes \(0.2 + 0.6 = 0.8\).
05

Simplify Denominator

Simplify the denominator. Calculate \(-0.9 - 0.7\), which equals \(-1.6\).
06

Compute the Slope

Substitute the simplified numerator and denominator into the formula: \[m = \frac{0.8}{-1.6} \]Simplify this fraction to get \(m = -0.5\).

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Key Concepts

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

Slope Formula
When discussing the slope of a line, we are referring to its steepness, described by the symbol \( m \). The slope formula is integral in understanding how the line trends over a set of points. The basic formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula essentially tells us the ratio of the vertical change (rise) to the horizontal change (run) between the two points. To use this formula effectively:
  • Make sure to subtract the \(y\)-coordinates (outputs) to calculate the rise.
  • Subtract the \(x\)-coordinates (inputs) to find the run.
  • Maintain the order of points to avoid errors: \((x_1,y_1)\) first, then \((x_2,y_2)\).
This consistent method will provide you the slope, a crucial component in graphing lines and understanding their direction.
Coordinate Geometry
Coordinate geometry, sometimes called analytic geometry, merges algebra and geometry through the use of a coordinate system. This system allows us to describe geometric shapes in a numerical way and calculate properties such as distances and slopes.In coordinate geometry:
  • A set of axes is used to define the position of points: typically the x-axis (horizontal) and y-axis (vertical).
  • Every point in the plane is denoted with a pair of numbers called coordinates \((x, y)\).
  • The algebraic representation allows for the precise calculation of geometric concepts like distances between points, midpoints, and the slope of a line.
By using coordinate geometry, complex geometric reasoning becomes a simpler set of algebraic problems, bridging two mathematical arenas.
Linear Equations
Linear equations are fundamental in algebra and represent relationships in a straight line when graphed on a coordinate plane. These equations typically take the form:\[ y = mx + b \]In this equation:
  • \(y\) and \(x\) represent the dependent and independent variables respectively.
  • \(m\) is the slope of the line indicating its steepness.
  • \(b\) is the y-intercept where the line crosses the y-axis.
Linear equations allow us to predict values and show the constant rate of change. Understanding linear equations in conjunction with slope provides a deeper insight into how variables interact linearly, paving the way for more complex analyses and applications.

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Most popular questions from this chapter

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. $$ f(x)=(x+4)^{2} $$

Graph function. \(f(x)=-4\)

Body Temperatures. The temperature in degrees Fahrenheit that is equivalent to a temperature in degrees Celsius is given by the linear function \(F(C)=\frac{9}{5} C+32 .\) Convert each temperature in the following excerpt from The Good Housekeeping Family Health and Medical Guide to degrees Fahrenheit. (Round to the nearest degree.) In disease, the temperature of the human body may vary from about \(32.2^{\circ} \mathrm{C}\) to \(43.3^{\circ}\) C for a time, but there is grave danger to life should it drop and remain below \(35^{\circ}\) C or rise and remain at or above \(41^{\circ} \mathrm{C}\).

The function $$T(a)=837.50+0.15(a-8,375)$$ (where \(a\) is adjusted gross income) is a model of the instructions given on the first line of the following tax rate Schedule X. a. Find \(T(25,000)\) and interpret the result. b. Write a function that models the second line on Schedule X.

Windchill. A combination of cold and wind makes a person feel colder than the actual temperature. The table shows what temperatures of \(35^{\circ} \mathrm{F}\) and \(15^{\circ} \mathrm{F}\) feel like when a 15 -mph wind is blowing. The relationship between the actual temperature and the windchill temperature can be modeled with a linear equation. a. Write the equation that models this relationship. Answer in slope-intercept form. b. What information is given by the \(y\) -intercept of the graph of the equation found in part (a)? $$ \begin{array}{|c|c|} \hline \multicolumn{2}{|c|}\text { 15-mph wind } \\ \hline \text { Actual temperature } \boldsymbol{x} & \text { Windchill temperature } \boldsymbol{y} \\ \hline 35^{\circ} \mathrm{F} & 25^{\circ} \mathrm{F} \\ \hline 15^{\circ} \mathrm{F} & 0^{\circ} \mathrm{F} \\ \hline \end{array} $$
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