Problem 31 In Problems 29-34, find an equat... [FREE SOLUTION]

91影视

Calculus Early Transcendentals: Pearson New International Edition

Dale Varberg, Edwin J. Purcell, Steve E. Rigdon

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 1: Problem 31

In Problems 29-34, find an equation for each line. Then write your answer in the form $A x+B y+C=0 .$ With $y$ -intercept 3 and slope 2

Short Answer

Expert verified

The equation in standard form is $2x - y - 3 = 0$.

Step by step solution

Identify Slope-Intercept Form

Given that the slope is 2 and the y-intercept is 3, we start by writing the slope-intercept form of the line, which is $y = mx + b$. Here, $m = 2$ and $b = 3$, so the equation is $y = 2x + 3$.

Convert to Standard Form

To convert the equation $y = 2x + 3$ into the standard form $Ax + By + C = 0$, we rearrange the terms. Subtract $2x$ from both sides to get $-2x + y = 3$.

Adjust to Desired Format

The standard form equation is typically written with positive $x$ coefficient. Multiply all terms in $-2x + y = 3$ by -1 to get $2x - y = -3$. Thus, in the form $A x+B y+C=0$, it is $2x - y - 3 = 0$.

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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 straightforward way to express the equation of a line. This form is written as $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

"$m$" or slope is the amount by which $y$ increases when $x$ increases by 1.
"$b$" or y-intercept is the value of $y$ when $x = 0$.

For example, in the equation $y = 2x + 3$:

Slope: $m = 2$, meaning for each unit increase in $x$, $y$ increases by 2 units.
Intercept: $b = 3$, meaning the line crosses the y-axis at $y = 3$.

This form is especially useful for graphing because it quickly shows how steep the line is and where exactly it crosses the y-axis.

Standard Form

The standard form of an equation of a line is expressed as $Ax + By + C = 0$ where $A$, $B$, and $C$ are integers. This form is commonly used in solving systems of linear equations or for certain algebraic manipulations.

In standard form, $A$ should be a non-negative integer.
Some people prefer the standard form because it doesn鈥檛 involve fractions or decimals as coefficients.

To convert from slope-intercept form such as $y = 2x + 3$ to standard form, follow these steps:

Rearrange it to $-2x + y = 3$.
Multiply all terms by -1 to ensure $A$ is positive, resulting in $2x - y - 3 = 0$.

Using the standard form can be beneficial in contexts where multiple equations need to be considered simultaneously.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the context of the slope-intercept form $y = mx + b$, the y-intercept is represented by $b$, which is a constant.Here鈥檚 why the y-intercept is important:

It provides a starting point on a graph where the line begins when $x = 0$.
It can help in graphing the line quickly because you know precisely where it intersects the y-axis.

In our example, the line equation is $y = 2x + 3$:

The y-intercept is $3$.
Thus, the line crosses the y-axis at the point $(0, 3)$.

Understanding the y-intercept helps in visualizing and sketching the graph of the line smoothly.

Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form $y = mx + b$, the slope is represented by $m$.

If $m > 0$, the line slopes upwards as you move from left to right.
If $m < 0$, the line slopes downwards.

The slope can be thought of as the rise over run:

"Rise" is the change in $y$.
"Run" is the change in $x$.

For the equation $y = 2x + 3$, the slope is 2:

With every one unit increase in $x$, $y$ increases by 2 units.
The slope determines how steep the line is and indicates how quickly $y$ changes with respect to $x$.

Understanding the slope is crucial for predicting how changes in one variable affect the other. This helps with plotting the line and seeing its general direction.

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91影视

In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) With \(y\) -intercept 3 and slope 2

Short Answer

Step by step solution

Identify Slope-Intercept Form

Convert to Standard Form

Adjust to Desired Format

Key Concepts

Slope-Intercept Form

Standard Form

Y-Intercept

Slope

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