Linear Equations: Finding Lines And Relationships

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Hey guys! Today, we're diving into the exciting world of linear equations. We'll tackle some problems involving finding the equation of a line given different pieces of information, and then we'll explore how to determine the relationship between pairs of linear equations. Let's get started!

1. Finding the Equation of a Line Through Two Points

So, our first task is to find the equation of a line that passes through two given points: A(4, -4) and B(2, 8). To do this, we'll need to use a couple of key concepts: the slope of a line and the point-slope form of a linear equation.

Calculating the Slope

The slope (m) of a line represents its steepness and direction. It's calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. The formula for the slope is:

m = (y2 - y1) / (x2 - x1)

In our case, (x1, y1) = (4, -4) and (x2, y2) = (2, 8). Plugging these values into the formula, we get:

m = (8 - (-4)) / (2 - 4) = 12 / -2 = -6

So, the slope of the line passing through points A and B is -6. That means for every one unit we move to the right along the x-axis, the line goes down six units along the y-axis. Knowing the slope is a crucial first step.

Using the Point-Slope Form

The point-slope form of a linear equation is a handy way to express the equation of a line when you know its slope and a point it passes through. The formula is:

y - y1 = m(x - x1)

Where:

  • (x1, y1) is a known point on the line
  • m is the slope of the line

We already know the slope (m = -6) and we have two points to choose from. Let's use point A (4, -4). Plugging these values into the point-slope form, we get:

y - (-4) = -6(x - 4)

Simplifying this, we have:

y + 4 = -6x + 24

Converting to Slope-Intercept Form

While the point-slope form is perfectly valid, it's often useful to convert the equation to slope-intercept form, which is:

y = mx + b

Where:

  • m is the slope
  • b is the y-intercept (the point where the line crosses the y-axis)

To convert our equation to slope-intercept form, we simply isolate y:

y = -6x + 24 - 4

y = -6x + 20

Therefore, the equation of the line passing through points A(4, -4) and B(2, 8) is y = -6x + 20. This tells us the line has a slope of -6 and crosses the y-axis at the point (0, 20).

2. Finding the Equation of a Line Given a Point and Slope

Now, let's tackle a slightly different problem. This time, we need to find the equation of a line that passes through the point (4, 5) and has a slope of 12. This is actually a bit easier than the previous problem because we're already given the slope.

Using the Point-Slope Form (Again!)

Since we have a point and a slope, the point-slope form is our best friend once again. Recall the formula:

y - y1 = m(x - x1)

In this case, (x1, y1) = (4, 5) and m = 12. Plugging these values into the formula, we get:

y - 5 = 12(x - 4)

Converting to Slope-Intercept Form

Again, let's convert this to slope-intercept form (y = mx + b) to make it more readable:

y - 5 = 12x - 48

y = 12x - 48 + 5

y = 12x - 43

So, the equation of the line passing through the point (4, 5) with a slope of 12 is y = 12x - 43. This line is much steeper than the previous one (slope of 12 versus -6) and crosses the y-axis at (0, -43).

3. Determining the Relationship Between Linear Equations

Alright, let's shift gears and talk about how to determine the relationship between two linear equations. Two lines can be related in one of three ways: they can be parallel, perpendicular, or neither (intersecting).

a. Analyzing Y = 2X - 8 and -2Y = -4X + 16

We have two equations:

  1. Y = 2X - 8
  2. -2Y = -4X + 16

To determine their relationship, let's rewrite the second equation in slope-intercept form (y = mx + b) by dividing both sides by -2:

Y = 2X - 8

Now, compare the two equations. Notice anything? They are identical! This means the two equations represent the same line. Therefore, they are neither parallel nor perpendicular; they are the same line. Another way to think about it is that the second equation is just a multiple of the first equation.

b. Analyzing 12X = 7 - Y and Y = -7 - 12X

We have two equations:

  1. 12X = 7 - Y
  2. Y = -7 - 12X

Let's rewrite the first equation in slope-intercept form. Add Y to both sides and subtract 12X from both sides:

Y = -12X + 7

Now we have:

  1. Y = -12X + 7
  2. Y = -12X - 7

Comparing these equations, we see that they have the same slope (-12) but different y-intercepts (7 and -7). This means the lines are parallel. Parallel lines have the same slope but never intersect.

Summary of Relationships

  • Same Line: The equations are identical (or multiples of each other). They have the same slope and the same y-intercept.
  • Parallel Lines: The equations have the same slope but different y-intercepts.
  • Intersecting Lines: The equations have different slopes. They intersect at a single point.
  • Perpendicular Lines: The equations have slopes that are negative reciprocals of each other (e.g., 2 and -1/2). Their product is -1.

Conclusion

And that's a wrap, folks! We've covered how to find the equation of a line given two points or a point and a slope, and we've learned how to determine the relationship between two linear equations. Understanding these concepts is fundamental to mastering linear algebra and other areas of mathematics. Keep practicing, and you'll become a pro in no time! Remember the point-slope form, the slope-intercept form, and how to calculate the slope. You got this!