Problem 57 - Introductory Algebra for College Students

Question

In Exercises \(57-64\), write an equation in the form \(y=m x+b\) of the line that is described. The \(y\) -intercept is 5 and the line is parallel to the line whose equation is \(3 x+y=6\).

Step-by-Step Solution

Answer

The equation of the line is \(y = -3x + 5\)

Step 1: Find the slope of the given line

First, convert the given equation, \(3x + y = 6\), to the slope-intercept form. This is done by moving \(3x\) term to the opposite side of the equation, which gives \(y = -3x + 6\). From this equation, it can be observed that the slope of the line is -3. This will be the slope of the line we are trying to find because it's parallel to this line.

Step 2: Write the equation of the line

From step 1, we have -3 as the slope (m) and from the problem statement, we have 5 as the y-intercept (b). We can now substitute these values into the slope-intercept form \(y = mx + b\) to get the equation of the line. This gives us \(y = -3x + 5\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the \( y \)-intercept. This format is particularly useful because it allows you to quickly identify important features of the line:

• The slope \( m \), which indicates how steep the line is and the direction it goes (upwards if positive, downwards if negative).
• The \( y \)-intercept \( b \), which shows where the line crosses the \( y \)-axis.

To convert any linear equation into the slope-intercept form, you simply solve for \( y \). For example, transforming the equation \( 3x + y = 6 \) to get \( y \) alone gives \( y = -3x + 6 \). This reveals that the line has a slope \( m \) of \(-3\) and a \( y \)-intercept \( b \) of \( 6 \).
Understanding this form allows you to quickly graph a line and determine its behavior.

Y-Intercept

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. It is a crucial part of the slope-intercept form \( y = mx + b \) because it provides a starting point for graphing the line. The \( y \)-intercept is denoted by \( b \).
In the equation \( y = mx + b \), when \( x = 0 \), the value of \( y \) is \( b \). This means that on a graph, the line will cross the vertical axis at this \( y\)-value. For instance, in the exercise where \( b = 5 \), the line crosses the \( y \)-axis at the point \((0, 5)\).
This concept helps in visualizing and drawing the line effectively, especially when you already know the slope. It also aids in solving real-world problems, as the \( y \)-intercept often represents a starting condition in many scenarios.

Parallel Lines

Parallel lines are lines in the same plane that never meet; they are always the same distance apart. For lines to be parallel, they must have identical slopes. This is because the slope indicates the angle at which the line rises or falls.
When two lines have the same slope but different \( y \)-intercepts, they are parallel but not coincident (they do not overlap). In our original exercise, the line \( 3x + y = 6 \) is converted to \( y = -3x + 6 \), giving it a slope of \(-3\). Any other line that also has a slope of \(-3\) will be parallel to this one.
Using this characteristic, we can determine that the line \( y = -3x + 5 \) is parallel to \( y = -3x + 6 \) because they both have a slope \(-3\). Understanding parallel lines helps in geometrical constructions and various applications like urban planning where parallel paths or roads are designed.