Problem 64 - Elementary Algebra

Question

How can you tell by inspection that \(y=2 x-4\) and \(y=-3 x-1\) are not parallel lines?

Step-by-Step Solution

Answer

The lines are not parallel because their slopes are different.

Step 1: Identify the Equation Forms

Both equations are in the slope-intercept form, which is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the Slope of Each Line

For the line equation \(y = 2x - 4\), the slope \(m_1\) is 2. For the line \(y = -3x - 1\), the slope \(m_2\) is -3.

Step 3: Compare the Slopes

Two lines are parallel if their slopes are equal. Here, \(m_1 = 2\) is not equal to \(m_2 = -3\).

Key Concepts

Slope-Intercept Form

In algebra, the slope-intercept form is a way of writing the equation of a line so you can easily identify its slope and y-intercept. This form is written as:

• \( y = mx + b \)
• \( m \) represents the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis

The main idea is that the slope \( m \) tells you how steep the line is. Meanwhile, \( b \) gives you the exact point where the line touches the y-axis.
For example, in the equation \( y = 2x - 4 \), the slope \( m \) is 2, and the y-intercept \( b \) is -4. This means the line rises 2 units for every 1 unit it moves to the right.

Equation Forms

Understanding different forms of an equation is crucial because they give you specific information about the line. The slope-intercept form is just one of several ways to express a linear equation. Here are a few others you might encounter:

• Point-slope form: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point on the line.
• Standard form: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.

Each form has its own uses, with slope-intercept being particularly handy for quickly determining the line's slope and y-intercept.
Choosing the right form can make solving, graphing, and understanding equations much easier.

Comparing Slopes

The slope of a line is a measure of its steepness, and it's what we compare to see if two lines are parallel. Remember, parallel lines never meet and have equal slopes.
To compare slopes:

• Look at the coefficient of \( x \) in each equation, which gives the slope.
• For example, in \( y = 2x - 4 \), the slope is 2.
• In \( y = -3x - 1 \), the slope is -3.
• If two slopes are equal, the lines are parallel.
• If they're different, like 2 and -3, the lines are not parallel.

In the given exercise, because the slopes are not equal, \( y = 2x - 4 \) and \( y = -3x - 1 \) are not parallel lines. This concept is crucial in geometry and helps us understand relationships between different lines.