Two Parallel Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the points of intersection of the following lines:

                        x + 2y = 6

                      2x + 4y = -9

Solution: 

Since the given equations are all first degree, then they are linear equations. They are straight lines. We can graph the two lines by getting their slope and y-intercept. 

For x + 2y = 6, 

                        x + 2y = 6

                              2y = - x + 6

                                y = -½ x + 3

                        slope (Δy/Δx), m = -½

                        y-intercept, b = 3

To trace the graph, plot 3 at the y-axis. This is your first point of the line (0, 3). Next, use the slope to get the second point. From the first point, count 2 units to the left and then 1 unit upward. 

For 2x + 4y = -9,

                        2x + 4y = -9

                               4y = - 2x - 9

                                 y = -½ x - ⁹/₄

                        slope (Δy/Δx), m = -½  

                        y-intercept, b = - ⁹/₄

To trace the graph, plot - ⁹/₄ at the y-axis. This is your first point of the line (0, - ⁹/₄). Next, use the slope to get the second point. From the first point, count 2 units to the left and then 1 unit upward.   

Photo by Math Principles in Everyday Life

From the graph, the two lines are parallel because their slopes are the same which is -½. The two lines will never meet how far they are extended. When you solve for x and y from the two given equations, their x and y will be equal to zero. From the two given equations,

                        x + 2y = 6

                      2x + 4y = -9

Multiply the first equation by 2 and -1 at the second equation. Add the two equations and let's see what will happen to x and y.

            2 (x + 2y = 6)                 2x + 4y = 12
                                       →
        - 1 (2x + 4y = -9)                -2x - 4y = 9
                                              ______________

                                                           0 ≠ 21

Since their x and y are equal to zero, then we cannot solve for x and y. Also, the right side of the final equation is not zero. Therefore, the two lines are parallel to each other.

Derivative - Chain Rule

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Given the following functions:

                      

                    
Find dy/dx in terms of x and y. 

Solution:  

The first thing that we have to do is to get the derivative of each functions with respect to their independent variables. 

For the first function

For the second function

For the third function

Therefore,

The above equation is called the Derivative by Chain Rule. As you noticed that the differentials like du and dv will be cancelled. Multiply the derivative of three functions, we have 

The final answer must be free from other variables like u and v. We have to eliminate u and v by substituting their values with x. We know that 

Therefore,

Solving Equations - Homogeneous Functions

Category: Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Find the general solution for the equation:

Solution:

The equation above is Differential Equation because it has dx and dy. The type of equation is Homogeneous because the functions and variables cannot be separated by Separation of Variables. There's a method to solve the Homogeneous Functions. Consider the given equation

Let   y = vx
      dy = vdx + xdv

Substitute y and dy to the given equation, we have

The above equation can now be separated by Separation of Variables. Divide both sides of the equation by x². 

Integrate both sides of the equation

but y = vx and v = ^y/_x, therefore,

Multiply both sides of the equation by 4x,