Linear Algebra Examples

Find the Domain (a-b)/(a+b+c)=(a-2b+c)/(a-b)
Step 1
Find the LCD of the terms in the equation.
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Step 1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 1.3
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 1.4
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 1.5
The factor for is itself.
occurs time.
Step 1.6
The factor for is itself.
occurs time.
Step 1.7
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Step 2
Multiply each term in by to eliminate the fractions.
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Step 2.1
Multiply each term in by .
Step 2.2
Simplify the left side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 2.2.2
Raise to the power of .
Step 2.2.3
Raise to the power of .
Step 2.2.4
Use the power rule to combine exponents.
Step 2.2.5
Add and .
Step 2.3
Simplify the right side.
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Step 2.3.1
Cancel the common factor of .
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Step 2.3.1.1
Factor out of .
Step 2.3.1.2
Cancel the common factor.
Step 2.3.1.3
Rewrite the expression.
Step 2.3.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 2.3.3
Simplify each term.
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Step 2.3.3.1
Multiply by .
Step 2.3.3.2
Multiply by by adding the exponents.
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Step 2.3.3.2.1
Move .
Step 2.3.3.2.2
Multiply by .
Step 2.3.3.3
Multiply by .
Step 2.3.4
Subtract from .
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Step 2.3.4.1
Move .
Step 2.3.4.2
Subtract from .
Step 2.3.5
Add and .
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Step 2.3.5.1
Reorder and .
Step 2.3.5.2
Add and .
Step 2.3.6
Add and .
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Step 2.3.6.1
Reorder and .
Step 2.3.6.2
Add and .
Step 3
Solve the equation.
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Step 3.1
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 3.2
Simplify .
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Step 3.2.1
Rewrite.
Step 3.2.2
Rewrite as .
Step 3.2.3
Expand using the FOIL Method.
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Step 3.2.3.1
Apply the distributive property.
Step 3.2.3.2
Apply the distributive property.
Step 3.2.3.3
Apply the distributive property.
Step 3.2.4
Simplify and combine like terms.
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Step 3.2.4.1
Simplify each term.
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Step 3.2.4.1.1
Multiply by .
Step 3.2.4.1.2
Rewrite using the commutative property of multiplication.
Step 3.2.4.1.3
Rewrite using the commutative property of multiplication.
Step 3.2.4.1.4
Multiply by by adding the exponents.
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Step 3.2.4.1.4.1
Move .
Step 3.2.4.1.4.2
Multiply by .
Step 3.2.4.1.5
Multiply by .
Step 3.2.4.1.6
Multiply by .
Step 3.2.4.2
Subtract from .
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Step 3.2.4.2.1
Move .
Step 3.2.4.2.2
Subtract from .
Step 3.3
Move all terms containing to the left side of the equation.
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Step 3.3.1
Subtract from both sides of the equation.
Step 3.3.2
Add to both sides of the equation.
Step 3.3.3
Combine the opposite terms in .
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Step 3.3.3.1
Subtract from .
Step 3.3.3.2
Add and .
Step 3.3.4
Add and .
Step 3.3.5
Multiply by .
Step 3.4
Move all terms not containing to the right side of the equation.
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Step 3.4.1
Add to both sides of the equation.
Step 3.4.2
Add to both sides of the equation.
Step 3.4.3
Subtract from both sides of the equation.
Step 3.4.4
Add and .
Step 3.5
Factor out of .
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Step 3.5.1
Factor out of .
Step 3.5.2
Factor out of .
Step 3.5.3
Factor out of .
Step 3.6
Divide each term in by and simplify.
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Step 3.6.1
Divide each term in by .
Step 3.6.2
Simplify the left side.
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Step 3.6.2.1
Cancel the common factor of .
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Step 3.6.2.1.1
Cancel the common factor.
Step 3.6.2.1.2
Divide by .
Step 3.6.3
Simplify the right side.
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Step 3.6.3.1
Move the negative in front of the fraction.
Step 3.6.3.2
Combine the numerators over the common denominator.
Step 3.6.3.3
Combine the numerators over the common denominator.
Step 4
Set the denominator in equal to to find where the expression is undefined.
Step 5
Subtract from both sides of the equation.
Step 6
The domain is all real numbers.
Interval Notation:
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