Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The equation in standard form is \(2x - y - 3 = 0\).

Step 1: 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 \(y = 2x + 3\).

Step 2: 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 \(-2x + y = 3\).

Step 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 \(2x - y - 3 = 0\).

Key Concepts

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’t 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’s 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.