Lines: Parallel, Perpendicular, Intersecting & Coincident

by Pedro Alvarez 58 views

Hey guys! Today, we're diving into the fascinating world of linear equations and how to determine the relative positions of pairs of lines. We'll not only figure out if they are parallel, secant, perpendicular, or coincident but also graph them on the Cartesian plane. So, grab your graph paper (or your favorite graphing tool) and let's get started!

Understanding Relative Positions of Lines

Before we jump into specific examples, let's quickly recap the different ways two lines can relate to each other in a plane:

  • Parallel Lines: These lines never intersect. They have the same slope but different y-intercepts. Think of railroad tracks – they run side by side without ever meeting.
  • Secant Lines: These lines intersect at a single point. Their slopes are different.
  • Perpendicular Lines: These are a special case of secant lines. They intersect at a right angle (90 degrees). The product of their slopes is -1.
  • Coincident Lines: These are essentially the same line. They have the same slope and the same y-intercept. If you graph them, you'll only see one line because they overlap perfectly.

Case a) y = 2x + 3 and y = 2x - 1

Let's start with our first pair of lines: y = 2x + 3 and y = 2x - 1. To determine their relative position, we need to look at their slopes and y-intercepts.

The equation of a line is generally written in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

For the first line, y = 2x + 3, the slope (m₁) is 2 and the y-intercept (b₁) is 3. For the second line, y = 2x - 1, the slope (m₂) is 2 and the y-intercept (b₂) is -1.

Notice anything interesting? Both lines have the same slope (m₁ = m₂ = 2) but different y-intercepts (b₁ = 3, b₂ = -1). This tells us that the lines are parallel. They have the same steepness but cross the y-axis at different points.

Graphing the Lines

To graph these lines, we can plot a couple of points for each. For y = 2x + 3:

  • If x = 0, then y = 2(0) + 3 = 3. So, we have the point (0, 3).
  • If x = 1, then y = 2(1) + 3 = 5. So, we have the point (1, 5).

For y = 2x - 1:

  • If x = 0, then y = 2(0) - 1 = -1. So, we have the point (0, -1).
  • If x = 1, then y = 2(1) - 1 = 1. So, we have the point (1, 1).

Plot these points on the Cartesian plane and draw a line through each pair of points. You'll see two lines running side by side, never intersecting, perfectly illustrating parallel lines.

In summary, the lines y = 2x + 3 and y = 2x - 1 are parallel.

Case b) y = x + 3 and y = -x - 2

Next up, we have the pair of lines: y = x + 3 and y = -x - 2. Let's break this down just like we did before.

For the first line, y = x + 3, the slope (m₁) is 1 (remember, if there's no number explicitly written before x, it's understood to be 1) and the y-intercept (b₁) is 3. For the second line, y = -x - 2, the slope (m₂) is -1 and the y-intercept (b₂) is -2.

Here, we see that the slopes are different (m₁ = 1, m₂ = -1). This means the lines will intersect, making them secant lines. But there's more to the story! Notice that the product of the slopes is 1 * (-1) = -1. This is the key indicator that the lines are also perpendicular.

Graphing the Lines

Let's graph these lines to visualize their relationship. For y = x + 3:

  • If x = 0, then y = 0 + 3 = 3. So, we have the point (0, 3).
  • If x = -3, then y = -3 + 3 = 0. So, we have the point (-3, 0).

For y = -x - 2:

  • If x = 0, then y = -0 - 2 = -2. So, we have the point (0, -2).
  • If x = -2, then y = -(-2) - 2 = 0. So, we have the point (-2, 0).

Plot these points and draw the lines. You'll observe that the lines intersect at a single point, forming a right angle. This confirms that the lines are indeed perpendicular and therefore also secant.

In conclusion, the lines y = x + 3 and y = -x - 2 are secant and perpendicular.

Case c) y = 3x + 2 and y = 3x + 2

Our final pair of lines is: y = 3x + 2 and y = 3x + 2. This one might seem a little too easy, but it's important to understand.

For the first line, y = 3x + 2, the slope (m₁) is 3 and the y-intercept (b₁) is 2. For the second line, y = 3x + 2, the slope (m₂) is 3 and the y-intercept (b₂) is 2.

Wait a minute... the slopes and the y-intercepts are exactly the same! (m₁ = m₂ = 3 and b₁ = b₂ = 2). This means that these are not two different lines, but rather the same line written twice. These lines are coincident.

Graphing the Lines

When you graph y = 3x + 2, you'll get a single line. If you try to graph the second equation, y = 3x + 2, it will fall perfectly on top of the first line. This is the hallmark of coincident lines.

To graph it, we can find two points:

  • If x = 0, then y = 3(0) + 2 = 2. So, we have the point (0, 2).
  • If x = 1, then y = 3(1) + 2 = 5. So, we have the point (1, 5).

Plot these points and draw the line. You'll see just one line, because the two equations represent the same line.

Therefore, the lines y = 3x + 2 and y = 3x + 2 are coincident.

Wrapping Up

And there you have it! We've successfully determined the relative positions of three pairs of lines and graphed them on the Cartesian plane. Remember, by comparing the slopes and y-intercepts, we can quickly identify whether lines are parallel, secant, perpendicular, or coincident. Graphing them helps visualize these relationships and solidify our understanding.

I hope this breakdown was helpful. Keep practicing, and you'll become a pro at identifying line relationships in no time! Happy graphing, everyone!