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Chapter 1: Problem 2

Find an equation for the line that passes through the given points. $$(0,0) and (1,1)$$

Short Answer

Expert verified
The equation of the line is \(y = x\).

Step by step solution

01

Understand the Equation of a Line

The equation of a line in slope-intercept form is given by \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept.
02

Find the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points \((0, 0)\) and \((1, 1)\) into the formula: \(m = \frac{1 - 0}{1 - 0} = 1\).
03

Determine the Y-intercept

Since one of the points given is the origin \((0,0)\), it directly gives us the y-intercept \(b = 0\).
04

Combine Slope and Y-intercept

With the slope \(m = 1\) and the y-intercept \(b = 0\), substitute these values into the slope-intercept form. This gives the equation \(y = 1x + 0\).
05

Simplify the Equation

The equation \(y = 1x + 0\) simplifies to \(y = x\). This is the equation of the line that passes through the given points.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is one of the most useful ways to express the equation of a straight line. This form is written as \(y = mx + b\).
Here, \(m\) represents the slope of the line, indicating its steepness and direction, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
  • If the slope \(m\) is positive, the line inclines upward as it moves from left to right.
  • If \(m\) is negative, it declines downward.
  • A line will be horizontal if \(m = 0\), indicating it is parallel to the x-axis.
  • The y-intercept is valuable because it allows you to visualize where the line intersects the y-axis.
Understanding this form makes it easier to quickly graph lines and understand their properties.
Slope Calculation
The slope of a line quantifies how steep it is. To calculate the slope, you don't need to guess how steep a line is by just looking at it; you can determine it precisely using a simple formula.
For two points \(x_1, y_1\)\ and \(x_2, y_2\), the slope \(m\) is calculated as follows:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
The idea is to find out how much the y-values change as the x-values change.
  • In our example, using points \((0,0)\) and \((1,1)\), we calculated the slope as \(m = 1\).
  • This tells us that for every unit we move right (along the x-axis), our y-value increases by the same amount.
  • It confirms a steady, linear increase in the values.
Mastering slope calculation helps in analyzing how two variables are related in uniform growth or decline.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis. In other words, it is the point at which the value of \(x\) is zero and only the y value is significant.
In the equation \(y = mx + b\), the y-intercept is represented by \(b\). One special situation is when one of your given points is the origin \(0, 0\), like in our example.
  • This automatically sets the y-intercept \(b\) to zero because the line passes through the origin.
  • Whenever the y-intercept is zero, it implies the line continues in its path directly from the origin without any upward or downward shift on the y-axis.
  • The y-intercept is crucial in graphical representation, helping you locate your line quickly on the graph.
Grasping the concept of y-intercepts allows you to understand the starting point of a line relative to the y-axis.

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Most popular questions from this chapter

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P=320\) when \(t=5\) and \(P=500\) when \(t=3\)

In 2010 , there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US. \(^{70}\) The number of vehicles grew \(15.5 \%\) over the previous decade, while the population has been growing at \(9.7 \%\) per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?

You have a budget of 2000 dollars for the year to cover your books and social outings. Books cost (on average) 80 each and social outings cost (on average) 20 dollars each. Let \(b\) denote the number of books purchased per year and \(s\) denote the number of social outings in a year. (a) What is the equation of your budget constraint? (b) Graph the budget constraint. (It does not matter which variable you put on which axis.) (c) Find the vertical and horizontal intercepts, and give a financial interpretation for each.

(a) What is the continuous percent growth rate for \(P=\) \(100 e^{0.06 t},\) with time, \(t,\) in years? (b) Write this function in the form \(P=P_{0} a^{t} .\) What is the annual percent growth rate?

Worldwide, wind energy " generating capacity, \(W,\) was 39,295 megawatts in 2003 and 120,903 megawatts in 2008 (a) Use the values given to write \(W\), in megawatts, as a linear function of \(t\), the number of years since 2003 . (b) Use the values given to write \(W\) as an exponential function of \(t\) (c) Graph the functions found in parts (a) and (b) on the same axes. Label the given values. (d) Use the functions found in parts (a) and (b) to predict the wind energy generated in \(2010 .\) The actual wind energy generated in 2010 was 196,653 megawatts. Comment on the results: Which estimate is closer to the actual value?
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