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Wednesday 18th of October
   
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Algebra
Order of Operations
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The Appearance of a Polynomial Equation
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Positive Integral Divisors
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Use of Parentheses or Brackets (The Distributive Law)
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Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
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Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
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Solving Rational Inequalities with a Sign Graph
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Solving an Equation with Two Radical Terms
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Intercepts of a Line
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Order of Operations
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Collecting Like Terms
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Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
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Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
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Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
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Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
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Algebra – Two Variables

This lesson begins to “solve for x and y” in problems of two equations and two unknowns.

System of Equations

Two or more equations that must all be true at the same time are called a system of equations . The values of the variables that make both equations true at the same time are the solution of a system.

y = 4x

x + y = 90

x = 18, y = 72
A system of equations. The solution of the system.

Many times the system of equations involves two equations and two unknowns. There are several methods to solve a system of equations. Some of the methods may seem familiar, and some may be new. They are all effective with two equations and two unknowns:

1. Guess and check

2. Solve a simpler problem

3. Draw a picture

4. Draw a graph

5. Adding equations

6. Variable substitution

 

Adding Equations

Algebra permits you to modify any equation, as long as you do the same thing to both sides, right? Believe it or not, this allows you to add two equations together.

The reasoning for adding two equations together goes like this. An equation is a statement of equality. The stuff on the left side equals the stuff on the right. The two sides are interchangeable. So when you add one equation to another, you are really adding the same amount (whatever unknown amount it is) to both sides.

Remember our analogy to a balance? Since both equations were in balance to begin with, the sum is still in balance. Although you may not know how many “pounds” you’re adding to both sides of the balance, you are adding the same number to both sides. It does not upset the balance; both sides remain equal.

The goal for adding equations is to eliminate one of the variables. So this method works when one equation has a variable that has the opposite value from the other equation. For example, if one equation contains “-7x” and the other contains “+7x” then adding the equations causes variable x to vanish, leaving you with one equation and one variable.

After you solve the remaining one equation for the one unknown, how do you solve for the other unknown? You can substitute the value into either one of the original two equations, and solve for the last unknown.

Example:

A + B = 50

A - B = 22

What are the values of A and B?

Solution:

Add both equations together: A + B = 50
+ A – B = 22
2A + B – B = 50 + 22
Combine similar terms: 2A = 72
Solve for A:  A = 36
Substitute back to find B: 36 + B = 50
  B = 50 – 36 = 18
Check the result with both: 36 + 18 = 50? Yes!
   36 – 18 = 22? Yes!
 
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