Question

Draw the graph of y = 2x – 1, y = 2x + 1 and y = 2x from x = 0 to x = 3. On the same graph paper and check whether these lines are parallel to each other.

Answer 1

Yes, these lines are parallel to each other because they all share the exact same slope m = 2.

To determine if lines are parallel, we examine their equations in the slope-intercept form, which is expressed as y = mx + c, where m represents the slope and c represents the y-intercept.

1. Identify the slopes

We compare the given equations to the standard form:

For y = 2x – 1, the slope m₁ = 2.

For y = 2x + 1, the slope m₂ = 2.

For y = 2x, the slope m₃ = 2.

2. Apply the parallelism condition

Two or more lines are defined as parallel if and only if their slopes are equal m₁ = m₂ = m₃ and their y-intercepts are different.

Slope condition: 2 = 2 = 2 (Satisfied)

Intercept condition: –1 ≠ 1 ≠ 0 (Satisfied)

3. Observe the geometric behavior

As shown in the graph from x = 0 to x = 3:

The lines maintain a constant distance from one another at every point.

They rise at the same rate 2 units up for every 1 unit right.

Because their steepness is identical, they will never intersect, regardless of how far they are extended.

The lines y = 2x – 1, y = 2x + 1 and y = 2x are parallel because they all have an identical slope of 2.