Problem 35 - Algebra Form and Function

Question

Are the lines parallel? $$ y=1+x ; y=1+2 x $$

Step-by-Step Solution

Answer

Answer: No, the lines are not parallel.

Step 1: Identify the slopes of the given lines

The given equations are in slope-intercept form (y = mx + b), where 'm' represents the slope of the line. So, for each equation, we can read the slope directly from the coefficients of the x terms. - For y=1+x, the slope m₁ is 1. - For y=1+2x, the slope m₂ is 2.

Step 2: Compare the slopes

Now that we have found the slopes of the two lines, we will compare them. If they are equal, the lines are parallel. - Since m₁ (1) is not equal to m₂ (2), the slopes are different.

Step 3: Conclusion

Because the slopes are different, we can conclude that the lines $$y = 1+x$$ and $$y = 1+2x$$ are not parallel.

Key Concepts

Understanding Slope-Intercept Form

To easily see the slope of a line, we often write the equation in slope-intercept form. This is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
Understanding this form is useful because it immediately gives us the slope and starting point of the line, making it easier to visualize. For instance, if we look at the equation \( y = 1 + x \):

• The slope \( m \) is 1, meaning the line rises one unit for every unit it moves horizontally.
• The y-intercept \( b \) is 1, indicating the line crosses the y-axis at (0, 1).

This form allows us to quickly sketch or analyze lines without needing further calculations.

Comparing Slopes

When determining if two lines are parallel, we look at their slopes. Parallel lines must have the same slope, so the key is to compare the slope values from the equations.
In our example, we have two equations:

• \( y = 1 + x \) with a slope of 1 (from the coefficient of \( x \))
• \( y = 1 + 2x \) with a slope of 2

By comparing these slopes, we see that 1 is not equal to 2. Therefore, these lines are not parallel. This comparison is crucial in many math problems to discern relationships between lines and is a direct application of understanding the slope-intercept form.

Drawing the Equation of a Line

Visualizing a line from its equation helps consolidate understanding of slope and intercept. To draw a line from its equation, start by using the slope-intercept form \( y = mx + b \). Identify the slope \( m \) and y-intercept \( b \) as your guides.

• Begin at the y-intercept point on the graph (at \( b \)).
• Use the slope \( m \) to determine the rise and run, moving vertically and horizontally to plot additional points.

For example, for \( y = 1 + x \), start at (0, 1). With a slope of 1, move one unit up and one unit right. Continue plotting these points and draw a line through them.
Similarly for \( y = 1 + 2x \), start at (0, 1) and move two units up for every unit across. Drawing these helps to confirm whether or not lines are parallel by observing their alignment.