System of Equations and Matrices

Let's talk about solving a system of equations using matrices, specifically Gaussian Elimination and Gauss-Jordan Elimination. In the first example, I will show the "in-between" steps using the row transformations. In the second example, I will not show the "in-between" steps, use the Gauss-Jordan Elimination method. Also it will a system of 4 equations with 4 unknowns.

Example 1: Solve using Gaussian Elimination

If you are able, always check your answer by plugging in what you found. If it is true for all, then it is a solution.


Example 2: Solve using Gauss-Jordan Elimination

First, convert the system of equations to a matrix. Then with each step, apply the row transformations. With the Gauss-Jordan method, you want to keep going until you get the identity matrix on the left-hand side. The identity matrix has a diagonal of 1's and the rest are zeroes. Gauss-Jordan Elimination is like Gaussian Elimination, but it goes a little further.

For more on matrices, check out my youtube channel. If you have any questions or blog ideas, you can e-mail me at jdmathguy@gmail.com

A Method for Factoring Polynomials: OI-Box Method

Let's talk about factoring polynomials. The method I'm going to show you is almost universal. It is a combination of the product-sum method, OI method, and the box method. This method of factoring can factor 2 terms (difference of squares), 3 terms (A=1), 3 terms (A doesn't =1), and 4 terms.

I call this method of factoring, the OI-box method. It is a variation on the OI method. If you are curious about the OI method, check out my recent YouTube video: OI method. It uses the Outer and Inner of FOIL. This method does the same. I'll explain the steps of the method first. Second, I will show a flowchart related to the method. Then I will show you some examples.

1. The Steps of Method

Step 1: Identify the number of terms and type.

Step 2: Apply the product-sum if necessary.

• Set-up. Put AC on the top and B on the bottom. 
• Find the left and the right by asking two questions
• What product equals AC?
• What sum equals B

Step 3: Apply the OI-box method. The solutions found in step 2 are the outer and inner or the O-I in FOIL.

• Set-up. Rows as O-I and Columns as F-L in FOIL
• Find the squares either by GCF of row and column or missing factor within a row or column.
• If needed, put negative only in the L-column.

Step 4: Take the diagonals found in step 3 and use them as factors.

Note: For trinomials (Ax²+Bx+C), they are factorable when B²-4AC equals a square (i.e. 0, 1,4,9,...). If not, it is prime.

2. Flowchart of the Method

3. 5 Examples using the Method

A. Factor polynomial with 4 terms

B. Factor trinomial (3 terms) with A equals 1

C. Factor trinomial (3 terms) with A doesn't equal 1

D. Factor perfect square trinomial (3 terms)

E. Factor a difference of squares (2 terms)

A. Example 1

B. Example 2

C. Example 3

D. Example 4

E. Example 5

Conclusion

For a method of factoring, it covers a wide variety of scenarios. The method cannot be used with a difference or sum of cubes, or when you have a difference of squares with 4 terms. Nonetheless this method of factoring can be very useful.

How Rational Expressions are like Fractions

Let's talk about rational expressions. When it comes to rational expressions, you need to know two things: factoring polynomials and fractions. It is important to know fractions because the steps for doing an operation with rational expressions is similar to an operation done between two fractions. Factoring is important because you can only cancel out factors with factors just like fractions.

In this post, I will compare the operations with fractions with rational expressions. I will line them up side by side. They should be the same if you set x equal to zero.

1. Simplifying rational expressions is like simplifying fractions

Note: you can only cancel out factors with factors.

2. Multiplying rational expressions is like multiplying fractions


Note: you can cancel within the same fraction or diagonally. It is always a factor of a numerator with the same factor in the denominator.

3. Dividing rational expressions is like dividing fractions





















Note: you have to first k-c-f it before you cancel things out. K-C-F stands for keep it, change it, and flip it.

4. Adding and subtracting rational expressions is like adding and subtracting fractions

Note: When you are adding and subtracting fractions, you have to find the LCD and equivalent fractions first to add the numerators. The denominators have to be the same. You don't cancel anything till its one fraction.

Simplifying rational expressions can be difficult, but remember they are just like fractions.

Deriving the Pythagorean Trigonometric Identities

Let's talk about the Pythagorean Trig Identities. Let's talk about how to derive the identities starting with Pythagorean theorem.

Using the Pythagorean theorem, we get:

Since sin(θ)=O/H and cos(θ)=A/H, then we get with some substitution:

Notice when you divide both sides of eqn. 1 by sin²θ.

Also see what happens when you take eqn. 1 and divide both sides by cos²θ.

So this is how you can derive the three Pythagorean Trigonometric Identities including…

Let's talk math.