Problem 51 - Algebra 1: Concepts and Skills

Question

Determine whether the graphs of the two equations are parallel lines. Explain your answer. $$line\quad a: y=x+8 line\quad b: x-y=-1$$

Step-by-Step Solution

Answer

Yes, the graphs of the two lines are indeed parallel as they have the same slope.

Step 1: Finding the slope of line a

To find the slope of line a \(y = x + 8\), you should recognize that this is already in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m_a\) of line a is 1.

Step 2: Finding the slope of line b

Unlike line a, line b \(x - y = -1\) is not immediately in slope-intercept form, so it needs to be rearranged. Subtracting \(x\) from both sides gives \(-y = -x -1\). Dividing through by -1 yields \(y = x + 1\). Here, the slope \(m_b\) of line b is also 1.

Step 3: Compare the slopes

The slopes \(m_a = 1\) and \(m_b = 1\) of line a and line b are clearly equal. Having equal slopes means that the lines are parallel.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when working with linear equations. It's written as \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the y-intercept, which is where the line crosses the y-axis. The slope provides information about the steepness and the direction of the line; a positive slope indicates an upward angle, while a negative slope signifies a downward angle.

For example, when the equation of a line is presented as \( y = 3x - 2 \), the slope (\( m \)) is 3 and the y-intercept (\( b \)) is -2. This format is particularly helpful for quickly sketching graphs and determining at a glance how the line will tilt and where it will cross the y-axis.

Using slope-intercept form simplifies the process of comparing lines, as you can easily spot their slopes and intercepts without needing additional calculations. This can save time and reduce the potential for errors when analyzing linear relationships.

Determining Slopes

The slope of a line is a measure of its steepness, typically represented as the ratio of the rise over the run between two points on the line. Essentially, it answers 'how much does y change for every unit increase in x?'.

To find a line's slope from an equation, you'll often want to manipulate it into the slope-intercept form if it isn't already. For line equations not in this form, such as \( ax + by = c \), you'll need to solve for \( y \) in terms of \( x \) to identify the slope.

Steps to Determine the Slope from an Equation:

• Get the equation in the form \( y = mx + b \) if it isn't already.
• Identify the number (coefficient) in front of \( x \), which is the slope \( m \).
• If there's no coefficient visibly written, the slope is 1 (as \( x \) is the same as \( 1x \)).

This procedure is precisely what was used in the solution to find the slope of line \( b \), which was not initially given in the slope-intercept form.

Comparing Slopes

When comparing slopes, you can determine the relationship between two lines. If two lines have the same slope, they are parallel and will never intersect. Contrastingly, if the slopes are negative reciprocals of each other (i.e., the product of the two slopes is -1), the lines are perpendicular and will intersect at a 90-degree angle.

In the exercise presented, we compared the slopes of two lines to see if they were parallel. Since both had a slope of 1, this meant they were indeed parallel. Always ensure you've correctly determined the slopes before comparing them; any mistakes in calculation can lead to incorrect conclusions about the relationship between the lines.

Being adept at comparing slopes is beneficial not only in geometry but also in real-life situations like understanding how changing one variable affects another in a proportional relationship, such as speed and travel time, or price and demand in economics.