Problem 100 - Algebra for College Students

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y-3 x=5,3 x+y=7$$

Step-by-Step Solution

Answer

The given lines are neither parallel nor perpendicular.

Step 1 - Rewrite each equation in slope-intercept form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with the first equation, \(y - 3x = 5\), add \(3x\) to both sides to isolate \(y\): \[ y = 3x + 5 \]Now do the same for the second equation, \(3x + y = 7\). Subtract \(3x\) from both sides: \[y = -3x + 7 \]

Step 2 - Identify the slopes of the lines

From Step 1, we have the equations in slope-intercept form: \(y = 3x + 5\) and \(y = -3x + 7\). Compare both equations to the slope-intercept form \(y = mx + b\) to identify the slopes. For the first equation, the slope \(m_1\) is 3. For the second equation, the slope \(m_2\) is -3.

Step 3 - Determine relationship using the slopes

Recall these rules:1. Two lines are parallel if their slopes are equal.2. Two lines are perpendicular if the product of their slopes is -1.3. If neither condition is true, the lines are neither parallel nor perpendicular. Using our values, \(m_1 = 3\) and \(m_2 = -3\), check: \[ m_1 = 3 eq -3 = m_2 \rightarrow \text{not parallel} \] \[ m_1 \times m_2 = 3 \times -3 = -9 eq -1 \rightarrow \text{not perpendicular} \]Thus, the given lines are neither parallel nor perpendicular.

Key Concepts

slope-intercept form

To understand the problem, we need to grasp the concept of **slope-intercept form**, which is a way of writing linear equations. This form is represented as: \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

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

Converting equations to slope-intercept form simplifies the process of comparing lines. For instance, consider the original equations from our exercise: \( y - 3x = 5 \) and \( 3x + y = 7 \).

First, we rearrange both to isolate \( y \). For \( y - 3x = 5 \):

Add \( 3x \) to both sides:
\[ y = 3x + 5 \]

For \( 3x + y = 7 \):
Subtract \( 3x \) from both sides:
\[ y = -3x + 7 \]

Now we have both in slope-intercept form, making it easier to identify and compare slopes.

slope

The **slope** of a line is crucial in understanding its direction and steepness. The slope is the \( m \) in the equation \( y = mx + b \). It represents the change in y for a one-unit change in x.

Positive slopes mean the line rises as it moves from left to right, while negative slopes indicate it falls.

In the exercise, we have two slopes:
For the equation \( y = 3x + 5 \), the slope \( m \) is 3.
For the equation \( y = -3x + 7 \), the slope \( m \) is -3.

With these values, we can quickly compare the lines' behaviors. Identical slopes indicate parallel lines, and slopes whose product is -1 suggest perpendicular lines. If neither is true, the lines are neither parallel nor perpendicular.

algebraic equations

Working with **algebraic equations** often involves manipulating them into more useful forms, such as the slope-intercept form.

Given equations often need to be rearranged. Basic algebraic operations like addition, subtraction, multiplication, and division allow us to isolate variables.

Let's revisit our original equations:
Starting with \( y - 3x = 5 \):
Adding \( 3x \) to both sides results in:

\[ y = 3x + 5 \]

For the second equation \( 3x + y = 7 \):
Subtracting \( 3x \) from both sides, we get:

\[ y = -3x + 7 \]

These steps demonstrate the power of algebraic methods in simplifying and solving equations.

Understanding the purpose of these operations helps us apply them effectively when determining the relationship between lines.