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Chapter 1: Problem 19

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 3 and \(y\) -intercept \((0,2)\)

Short Answer

Expert verified
The equation is \(3x - y = -2\).

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 3 \) and \( b = 2 \).
02

Substitute the Values

Plug the given slope and y-intercept into the slope-intercept form equation. This gives us \( y = 3x + 2 \).
03

Rearrange to Standard Form

The standard form of a line's equation is \( Ax + By = C \). To convert, subtract \( y \) from both sides of \( y = 3x + 2 \) to get \( 0 = 3x - y + 2 \).
04

Simplify the Equation

Rearrange the equation to align with \( Ax + By = C \), resulting in \( 3x - y = -2 \). Now the equation is in standard form.

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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 a way to express the equation of a line simply and directly. It is written as \( y = mx + b \). Here, \( y \) represents the dependent variable or output, \( m \) is the slope of the line, \( x \) is the independent variable or input, and \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis.
  • The slope \( m \) describes how steep the line is and in which direction it tilts. If \( m \) is positive, the line rises as it moves left to right. If negative, it falls.
  • The y-intercept \( b \) is the starting value of \( y \) when \( x \) is zero. It's the point \((0, b)\) on the graph.
Knowing \( m \) and \( b \), you can immediately sketch out the line. This form makes it easy to identify the characteristics of the line quickly.
Standard Form
Standard Form is another way to express the equation of a line. It is framed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative number. Here鈥檚 what you should know:
  • This form is a bit more flexible when dealing with nondiscrete intersections or varied line relations.
  • Standard form is often preferred for solving systems of linear equations and finding intersections since you can manipulate equations more easily.
  • It requires a specific setup: typically, \( A \), \( B \), and \( C \) are kept to their simplest integer ratio.
Transforming from slope-intercept to standard form may involve rearranging and simplifying the terms, ultimately illustrating the relationship between \( x \) and \( y \) differently.
Equation of a Line
The equation of a line connects linear algebra to geometry by describing a straight line in a 2-dimensional space. Using the slope-intercept or standard form, you essentially convey the same linear relationship; they just offer different insights.
  • In the slope-intercept form, emphasis is on the slope and starting point \( (0, b) \). It鈥檚 great for straightforward graphing and immediate analysis of how the line behaves.
  • By contrast, the standard form handles algebraic manipulation better, especially with systems of equations.
  • Both formats can be transformed into one another wherein the slope \( m \) is computed as \( -\frac{A}{B} \) from standard form, and vice versa for conversions.
Equations of a line offer a versatile tool set for representing, analyzing, and solving real-world problems using theoretical and graphical insights.

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Most popular questions from this chapter

Daphnia is a genus of zooplankton that comprises a number of species. The body growth rate of Daphnia depends on food concentration. A minimum food concentration is required for growth: Below this level, the growth rate is negative; above, it is positive. In a study by Gliwicz (1990), it was found that growth rate is an increasing function of food concentration and that the minimum food concentration required for growth decreases with increasing size of the animal. Sketch two graphs in the same coordinate system, one for a large and one for a small Daphnia species, that illustrates this situation.

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

When log \(y\) is graphed as a function of log.x, a straight line results. Graph straight lines, each given by two points, on a log-log plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(4,2),\left(x_{2}, y_{2}\right)=(8,8) $$

Yang et al. (2014) measured the time that animals of different sizes spend urinating. For animals larger than \(1 \mathrm{~kg}\), the time spent urinating increases (slowly) with animal size. The smallest animal in their study was a cat (mass \(5 \mathrm{~kg}\), duration of urination 18s) and the largest was an elephant (mass \(5000 \mathrm{~kg}\), duration of urination \(29 \mathrm{~s}\) ). Make a sketch of time spent urinating as a function of animal size.

When \(\log y\) is graphed as a function of \(x\), a straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(0,5),\left(x_{2}, y_{2}\right)=(3,1) $$
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