Solving Quadratic Equations of Two Unknowns, 2

Category: Algebra

"Published in Newark, California, USA"

Solve the following systems by substitution:

Solution:

Consider the given equations above 

Rewrite the second equation in terms of y at the left side as follows

Substitute the value of y to the first equation, we have

By using quadratic formula, the values of x are

 

 

 

If you will choose the positive sign, then the value of x is

Substitute the value of x to the second equation in order to get the value of y, we have

 

If you will choose the negative sign, then the value of x is

Substitute the value of x to the second equation in order to get the value of y, we have 

Therefore, the solutions of the two equations are:  

 

Solving Quadratic Equations of Two Unknowns

Category: Algebra

"Published in Newark, California, USA"

Solve the following systems by elimination:

Solution:

Consider the given equations above

Multiply the first equation by 2 and the second equation by -1, we have

Add the two equations in order to solve for the value of y.

----------------------------

Substitute the value of y to either of the two equations in order to solve for the value of x, we have

Therefore, the solutions of the two equations are: 

 

Rotation of a Circle, 2

Category: Analytic Geometry

"Published in Newark, California, USA"

Given the equation of a circle

Find the new equation of a circle if the given circle is rotated counterclockwise about the origin at 30°.  

Solution:

The first thing that we need to do is to write the given equation in standard form as follows

From the standard form, we can draw the circle as follows

Photo by Math Principles in Everyday Life

If the given equation is written in rectangular coordinate system, then we need to convert it into polar coordinate system as follows 

    

Next, substitute θ with θ - 30° and then expand using the sum and difference of two angles formula, we have 

Convert the above equation into rectangular coordinate system in order to get its final equation. Therefore, the new equation of a circle is