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Rotating a Parabola
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Mixed Numbers and Improper Fractions
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Literal Numbers
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Simplifying Complex Fractions
Decimals and Fractions
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Logarithmic Functions
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The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
Polynomials
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Numbers
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
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
Dividing Radical Expressions
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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Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Exponents
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
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
Monomial Factors
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
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
   
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Solving Linear Equations

In this section we will combine all of the methods that we have learned to solve any linear equation containing fractions, decimals, and parentheses. We will use all of properties to simplify the expressions on both sides of the equation.

DEFINITION A simple linear equation in one variable is any equation that can be put in the form of ax + b = 0, where a and b are real numbers, a ≠ 0.

The alternate form of this equation is: Ax = B or A = Bx

Extend this to the general form of the equation.

General equation: Ax + B = Cx + D (A, B, C, D are real numbers.).

i) Simplify the equation so that there are only integers on both sides

ii) Many equations have enclosures which must first be simplified.

iii) Then solve simplified equations vertically - using the balance beam.

Pattern: ax + b = cx + d Both sides simplified (a ,b, c, d are integers.) Look at the coefficients of x and determine which is the larger integer (furthest to the right on the number line). If c > a then we will keep the variable x on that side of the equation and keep the constant on the other side. To do this we first add opposites on the balance beam below the equation. Look at the pattern, and then follow the same steps through several examples

Solve simplified equations vertically - using the balance beam .

Equation: a x + b = c x + d Both sides simplified (a ,b, c, d are integers.)

Pattern: c > a

1) Add opps:

Complete the step: (b - d) = (c - a)x → Let A = (c - a) and B = (b - d)

Then B = Ax → A > 1 is coefficient of x

2) Multiply recip:

Example 1:

Solve 3(x − 2) = 5P

rocedure: 3(x − 2) = 5

1) Remove parentheses: 3x − 6 = 5

2) Add opps:

Complete the step:

Note: (5 + 6) = 11

Then 3x = 11     3 is the coefficient of x

3) Multiply recip: Since and

Then

4) Check in original equation: 3(x − 2) = 5

Replace in the equation:

Simplify equations that contain decimal fractions first by multiplying by the LCD.

Example 3:

Solve 2(x − 3) + (x − 4) = − 5 (x +1) − 7(2x – 3)

Procedure: 2(x − 3) + (x − 4) = - 5(x +1) − 7(2x – 3)

Remove parentheses: 2x − 6 + x − 4 = - 5x − 5 − 14x + 21 Distributive Property

Collect like terms: 3x − 10 = -19x + 16 Commutative Property

Simplify using balance beam:

1) Add opps: Add same thing to both sides
   
Complete the step: (3 + 19)x = +10 (3 + 19) = 22 and (16+10) = 26
Then 22x = 26 22 is the coefficient of x
2) Multiply recip: Since

Then

3) Check in original equation: 2(x − 3) + (x − 4) = - 5(x +1) − 7(2x – 3)

Replace x

 
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