Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(4 x+5 y=-20\)

Step-by-Step Solution

Answer

Slope: \(-\frac{4}{5}\), y-intercept: \(-4\).

Step 1: Identify the standard form

The given equation is in standard form: \(4x + 5y = -20\). In this form, \(Ax + By = C\), where \(A = 4\), \(B = 5\), and \(C = -20\).

Step 2: Convert to slope-intercept form

To find the slope and y-intercept, we must convert the equation to slope-intercept form \(y = mx + b\). Start by isolating \(y\): Subtract \(4x\) from both sides: \(5y = -4x - 20\).

Step 3: Solve for y

Divide every term by 5 to solve for \(y\): \(y = \frac{-4}{5}x - 4\).

Step 4: Identify the slope and y-intercept

In the equation \(y = \frac{-4}{5}x - 4\), the slope \(m\) is \(-\frac{4}{5}\), and the y-intercept \(b\) is \(-4\).

Key Concepts

Standard Form of a Line

Linear equations can often be seen in different forms, and one popular representation is the Standard Form of a Line. This is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. It gives a straightforward way of looking at a line, but it doesn't immediately reveal the slope or y-intercept, which are vital for understanding the line's behavior.
To use this form effectively:

• The coefficients \(A\), \(B\), and \(C\) should ideally be integers.
• The equation represents a straight line in 2D space.
• Manipulating the form, such as changing it to slope-intercept form, reveals important characteristics like slope and y-intercept.

Given the equation \(4x + 5y = -20\), you'll see it's in standard form, but notice, to find the slope and y-intercept, you'll need to convert this.

Slope-Intercept Form

The Slope-Intercept Form of a linear equation is one of the most user-friendly and intuitive ways to understand the dynamics of a line. In the form \(y = mx + b\), we can instantly identify:

• The slope \(m\), which tells us how steep the line is.
• The y-intercept \(b\), which tells us where the line crosses the y-axis.

This form makes it easy to graph a line since knowing \(m\) and \(b\) allows you to plot the y-intercept and follow the slope to draw the line.
For instance, after converting \(4x + 5y = -20\) to the form \(y = \frac{-4}{5}x - 4\), you immediately see the slope is \(-\frac{4}{5}\), and the y-intercept is \(-4\). Thus, the slope-intercept form empowers you to quickly analyze and graph the line without complex calculations.

Linear Equations

Linear equations represent straight lines and are foundational in algebra and geometry. They graph as straight lines and are categorized by their constant change rate, known as the slope. Linear equations can take multiple forms, with standard form and slope-intercept form being the most common.
Once a linear equation is defined, it can describe real-world relationships, such as speed over time or cost depending on quantity. To work effectively with linear equations:

• Familiarize yourself with standard and slope-intercept forms.
• Understand the significance of slope and y-intercept in any context.
• Develop skills to seamlessly convert between forms to reveal different aspects of the line.

In our exercise, we start with \(4x + 5y = -20\), a typical linear equation in standard form, and convert it to better understand its characteristics through the slope and y-intercept.

Isolating Variables

Isolating variables is a crucial algebraic skill, enabling us to solve equations and understand their components. When converting from Standard Form, \(Ax + By = C\), to Slope-Intercept Form, \(y = mx + b\), isolating the variable \(y\) is key.
This involves algebraic operations like addition, subtraction, multiplication, and division to "isolate" the desired term. To convert \(4x + 5y = -20\):

• Subtract \(4x\) from both sides: \(5y = -4x - 20\).
• Divide every term by 5 to solve for \(y\): \(y = \frac{-4}{5}x - 4\).

By mastering these steps, you can confidently transform equations to reveal valuable insights such as the slope and y-intercept, which give clarity to the equation's geometric representation.