Showing posts with label combine like terms. Show all posts
Showing posts with label combine like terms. Show all posts

Solving Linear Equations: Part II

In this section we will solve more complicated linear equations.

If you can, combine same side like terms first then opposite side like terms using the techniques that we have learned in part I. Caution: A common error is to add or subtract a term on the same side of the equal sign.  Only use the opposite operation on opposite sides of the equal sign.

Not all equations work out to have a single solution.  Some have infinitely many solutions such as x = x.  Here any number the we chose for x will produce a true statement.  Also, some equations have no solution such as x + 1 = x.

   Contradiction – An equation that will always be false has no solution.
   Identity – An equation that will always be true has any real number, R, as a solution.
It is quite common to encounter linear equations that require us to distribute before combining like terms.

Literal equations are difficult for many people because there will be more than one variable.  Just remember that the letters are place holders for some number, so the steps for solving are the same.  Isolate the variable that it asks us to solve for.

Word Problem: The internet connection at the hotel costs $5.00 to log on and $0.20 a minute to access.  If Joe's total bill came to $18.00 then how many minutes did Joe spend on the internet? (Set up an algebraic equation and solve it)

Video Examples on YouTube:

Multiplying Polynomials

The distributive property and the product rule for exponents are the keys to multiplication of polynomials.

Remember the product rule,
It says that when we multiply two numbers with the same base we must add the exponents.

Monomial x Polynomial
A common mistake is to distribute when working with multiplication. The distributive property only applies when addition or subtraction separates the terms.
Binomial x Polynomial: Use the distributive property to multiply polynomials by binomials.
We can save a step by distributing each term in the binomial individually. Sometimes this technique is referred to as the FOIL method. (FOIL – Multiply the First, Outer, Inner and then Last terms together.)

When combining like terms, be sure that the variable parts are exactly the same. Go slow and work in an organized fashion because it is easy to make an error when many terms are involved.
It is good practice to recheck your distributive step.
Trinomial x Polynomial
The following problem is one that we will have to be able to do before moving on to the next algebra course. These are tedious, time consuming, and often worked incorrectly. Use caution, take your time and work slowly.
Function notation for multiplication looks like
Given the functions f and g find (f g)(x).
The following special products will simplify things if we memorize them.
This special product is often called difference of squares. Notice that the middle terms cancel because one term will always be positive and the other will be negative.
We may use these formulas as templates when multiplying binomials. It is a good idea to memorize them.

Video Examples on YouTube: