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Adding Fractions
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The FOIL Method
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Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
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Multiplying and Dividing in Scientific Notation
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Multiplication by 111
Adding Fractions
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Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
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Complex Numbers
Factoring the Difference of Two Squares
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Factoring Out the Opposite of the GCF
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Properties of Exponents
Scientific Notation
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Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
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Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
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Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Solving Systems of Equations - Two Lines

 It is important to note that some equations have decimals or fractions as coefficients. Systems do not always have nice integers in the solutions

Example 1:

When the equations have fraction coefficients we must multiply each equation by its LCM to obtain integral coefficients.

  and  find 

Now we must multiply each equation so that one of the variables has coefficients that are “equal and opposite”.

  now find

Add the equations then simplify to get x = 56

Replace x = 56 in the equation: 7(56) − 6y = 2

392 − 6y = 2 Add opps: - 6y = - 390

multiply recip: to find y = 65

Check the values in both equations The solution point: { (56, 65) }


Addition Method:

Add two equations in a system of equations, and obtain another equation in the system (having the same solution). Also multiply an equation by a real number and obtain another equation in the system, and combine the two processes.

The object in the Addition Method is to add two of the equations in order to eliminate one of the variables. The resulting equation can then be solved for either x = h or y = k and which can then be used as a replacement in one of the given equations to find the value of the other one.


Example 2:

Multiply the first equation by -4

Now add and get

This gives

Replace in the first equation:

Check the values in both equations

The solution point:


Example 3:

(1) See that the LCM of the 1st equation is 12 and of the 2nd is 6.

(2) Multiply each by the LCM’s

(3) Now solve the simplified system --.

  To get equal and opposites multiply the 1st by 2 and the 2nd by +3 then add the results:

Replace x = 2 in (3): 4(2) + 3y = 14 which yields y = 2

Check: Always check your answers.

Go back to the original equations and substitute both: 

They checked -- now write the solution as S = { (2, 2) }

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