How this equation represent the equation of line?

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In summary, the equation of a line that includes the point (x, y, z) can be found by solving the equation (x-2/3, y-6/9, z-9/5) = 0.
  • #1
Hyperspace2
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Hello people. In my textbooks the question is asked as following
1. Find the S.D(short distance)between the lines
a) x-2/3=y-6/9=z-9/5
b) 5x + 2y-6z-5 = 0 = 4x+6y+7z-8
My problem is not about of solution.

Solution is preety well given in book. But
My main question is
how equation b represent the equation of line


Further,it states

5x + 2y-6z-5 + K(4x+6y+7z-8) = 0 represents the equation of plane which consist line.
How could this be?
 
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  • #2
Hello Hyperspace2! :smile:
Hyperspace2 said:
b) 5x + 2y-6z-5 = 0 = 4x+6y+7z-8

My main question is
how equation b represent the equation of line

For example, x = y = 0 is obviously a line (the z-axis).

Similarly, any two independent linear equations make a line.

This is because anyone linear equation makes a plane, so two together makes all the points on both planes, ie the intersection, which is a line! :wink:
5x + 2y-6z-5 + K(4x+6y+7z-8) = 0 represents the equation of plane which consist line.
How could this be?

It means that, for any fixed value of K, that is the equation of a plane which includes line b.

It's true because if you put any point on line b into that equation, it becomes 0 + K0 = 0 … which is obviously true! :smile:
 
  • #3
tiny-tim said:
For example, x = y = 0 is obviously a line (the z-axis).


tiny-tim said:
.

This is because anyone linear equation makes a plane, so two together makes all the points on both planes, ie the intersection, which is a line! :wink:


It means that, for any fixed value of K, that is the equation of a plane which includes line b. :
tiny-tim said:
.
It's true because if you put any point on line b into that equation, it becomes 0 + K0 = 0 … which is obviously true! :smile:

Thanks I never thought of that . I am crystal clear now.
 
  • #4
Another way of looking at it is this:
The "compound" equation (x-2)/3= (y-6)/9= (z-9)/5 (I presume you intended those parentheses) can be broken into two equations, (x- 2)/3= (y- 6)/9 and (y- 6)/9= (z- 9)/5 which are the equations of two planes. The two equations, together, are satisfied by points that lie on the intersection of the two planes, a line.

Finally, Since those three fractions are equal to one another, set them all equal to "t":
(x- 2)/3= t so x- 2= 3t and x= 3t+ 2,
(y- 6)/9= t so y- 6= 9t and y= 9t+ 6
(z- 9)/6= t so z- 9= 6t and z= 6t+ 9

Those are three linear parametric equations and so are parametric equations of a line.
 
  • #5
HallsofIvy said:
Another way of looking at it is this:
The "compound" equation (x-2)/3= (y-6)/9= (z-9)/5 (I presume you intended those parentheses) can be broken into two equations, (x- 2)/3= (y- 6)/9 and (y- 6)/9= (z- 9)/5 which are the equations of two planes. The two equations, together, are satisfied by points that lie on the intersection of the two planes, a line.

Finally, Since those three fractions are equal to one another, set them all equal to "t":
(x- 2)/3= t so x- 2= 3t and x= 3t+ 2,
(y- 6)/9= t so y- 6= 9t and y= 9t+ 6
(z- 9)/6= t so z- 9= 6t and z= 6t+ 9

Those are three linear parametric equations and so are parametric equations of a line.

Thanks sir for making a more clear explanation.
 

FAQ: How this equation represent the equation of line?

1. What is the equation of a line?

The equation of a line is a mathematical representation of a straight line on a graph. It is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept, the point where the line intersects the y-axis.

2. How does the equation y = mx + b represent a line?

The equation y = mx + b represents a line because it follows the slope-intercept form, which is used to graph lines. The coefficient m represents the slope of the line, and the constant b represents the y-intercept. By plugging in different values for x, we can find the corresponding y values and plot them on a graph, creating a line.

3. What does the slope of a line represent?

The slope of a line represents the rate of change between two points on the line. It can also be thought of as the steepness of the line. A positive slope indicates an upward direction, while a negative slope indicates a downward direction.

4. How is the slope calculated in the equation y = mx + b?

The slope in the equation y = mx + b is represented by the coefficient m. It is calculated by taking the change in the y-values divided by the change in the x-values between two points on the line. In other words, it is the rise over the run.

5. Why is the y-intercept important in the equation of a line?

The y-intercept is important because it tells us the point where the line intersects the y-axis. This point is essential for plotting the line on a graph and understanding the behavior of the line. It can also tell us the initial value of the dependent variable y when x is equal to 0.

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