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Elimination Using Multiplication
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Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
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Adding and Subtracting Fractions

Examples with solutions:

Example 1:

Perform the addition , and express your final answer in simplest form.

solution:

The prime factorizations of the two denominators are

10 = 2 1 × 5 1

15 = 3 1 × 5 1

So, the prime factors 2, 3, and 5 occur, each to at most the first power. Thus

LCD = 2 × 3 × 5 = 30

(Notice that this is smaller than the product, 10 × 15 = 150, of the original denominators. The factor 5 occurs in both of the original denominators, but need appear only once in the LCD.)

Now, to convert 3 / 10 to an equivalent fraction with a denominator of 30, we need to multiply top and bottom by 30 / 10 = 3. To convert 7 / 15 to an equivalent fraction with denominator of 30, we need to multiply top and bottom by 30 / 15 = 2. So

Since 23 is a prime number, no simplification of this result is possible, and so our final answer is

 

Example 2:

Perform the addition and express your final answer in simplest form. {\b

solution:

First we write the two denominators as products of prime factors:

48 = 2 4 × 3 1

18 = 2 1 × 3 2

Thus,

LCD = 2 x · 3 y

since the prime factorizations of 48 and 18 contain only 2 and 3 as factors.

Then  
  x = 4, because the highest power of 2 is 4, occurring in the factorization of 48,
and  
  y = 2, because the highest power of 3 is 2, occurring in the factorization of 18.

So,

LCD = 2 4 · 3 2 = 144.

Now, to convert 25 / 48 to an equivalent fraction with a denominator of 144, we must multiply top and bottom by 144 / 48 = 3. To convert 7 / 18 to an equivalent fraction with a denominator of 144, we must multiply top and bottom by 144 / 18 = 8. So, our problem becomes

To check for the possibility of simplification, we need to express the numerator and denominator of this result as a product of prime factors. We already know that

144 = 2 4 · 3 2

It takes just a minute to verify that 131 is not divisible by either 2, 3, 5, 7, or 11, and so 131 must be a prime number already. Therefore no further simplification is possible and our final answer is

(In case you’re wondering why we had to check only that none of 2, 3, 5, 7, and 11 divided evening into 131 to conclude that 131 is a prime number – the reason is this. We don’t have to check any divisors which are not prime numbers themselves, or which are larger than the square root of the number being factored. Since the square root of 131 is less than 12, we only have to check potential prime divisors which are less than 12, and this is the list 2, 3, 5, 7, and 11.)

 

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