Problem 41 - Algebra 1: Concepts and Skills

Question

Determine whether the graphs of the two equations are parallel lines. Explain. $$ \begin{aligned} &\text { line a: } 3 x+9 y+2=0\\\ &\text { line } b: 2 y=-6 x+3 \end{aligned} $$

Step-by-Step Solution

Answer

No, the lines are not parallel.

Step 1: Writing line a in slope-intercept form

First, transform the equation 3x + 9y + 2 = 0 to the slope-intercept form. Start by isolating y: 9y = -3x - 2, then divide every term by 9 to get \(y = -\frac{1}{3}x - \frac{2}{9}\).

Step 2: Writing line b in slope-intercept form

Now, do the same for the line b equation: 2y = -6x + 3; divide every term by 2 to get \(y = -3x + \frac{3}{2}\).

Step 3: Comparing slopes

After transforming both equations, it's observable that the first line has a slope of -1/3 and the second one, a slope of -3. Since the two slopes aren't equal, it can be concluded that the lines are not parallel.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to express the equation of a line. It is written as:

• \( y = mx + b \)

In this equation, \( m \) represents the slope, and \( b \) signifies the y-intercept. The slope (\( m \) indicates how steep the line is and which direction it goes:

When \( m \) is positive, the line ascends.

When \( m \) is negative, the line descends.

An \( m \) of zero denotes a horizontal line.To find the slope-intercept form from other forms, you need to isolate \( y \) on one side of the equation to see the slope and the intercept clearly. This format eases graphing and understanding the behavior of a line, as we can immediately determine both the slope and y-intercept by simple inspection.

Equation of a Line

The equation of a line can be presented in different forms, but they all essentially describe straight lines on a coordinate plane. The slope-intercept and the standard form are among the most commonly used.

• Standard form is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integer values, and both \( A \) and \( B \) are not zero.
• The slope-intercept form, previously covered, is more intuitive for graphing.

Converting between these forms often involves algebraic manipulation such as isolating variables or rearranging terms. For example, converting from standard form to slope-intercept form requires you to solve for \( y \). However, no matter the starting point, the main goal is to express the relationship between \( x \)-coordinates and \( y \)-coordinates that the line passes through.

Comparing Slopes

Comparing the slopes of two lines is essential to determine their relationship on a graph. Specifically:

• Parallel lines have identical slopes.
• Perpendicular lines have slopes that are negative reciprocals.
• If two lines have different slopes, they will intersect at some point and are neither parallel nor perpendicular.

When given two equations, convert them to the slope-intercept form to easily identify the slopes. In cases where the slopes are equivalent, it is a sign that the lines are parallel.
For example, if lines have slopes like \( -\frac{1}{3} \) and \( -3 \), they clearly differ, indicating they are not parallel. Parallel line identification aids in many areas of geometry and algebra, impacting how shapes and graphs are understood in broader mathematical contexts.