Finite Math Examples

Solve for z 2/z-7/(z-4)=1/3
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
Since contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for are:
1. Find the LCM for the numeric part .
2. Find the LCM for the variable part .
3. Find the LCM for the compound variable part .
4. Multiply each LCM together.
Step 1.3
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 1.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 1.5
Since has no factors besides and .
is a prime number
Step 1.6
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 1.7
The factor for is itself.
occurs time.
Step 1.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 1.9
The factor for is itself.
occurs time.
Step 1.10
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Step 1.11
The Least Common Multiple of some numbers is the smallest number that the numbers are factors of.
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
Simplify each term.
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Step 2.2.1.1
Rewrite using the commutative property of multiplication.
Step 2.2.1.2
Multiply .
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Step 2.2.1.2.1
Combine and .
Step 2.2.1.2.2
Multiply by .
Step 2.2.1.3
Cancel the common factor of .
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Step 2.2.1.3.1
Cancel the common factor.
Step 2.2.1.3.2
Rewrite the expression.
Step 2.2.1.4
Apply the distributive property.
Step 2.2.1.5
Multiply by .
Step 2.2.1.6
Cancel the common factor of .
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Step 2.2.1.6.1
Move the leading negative in into the numerator.
Step 2.2.1.6.2
Factor out of .
Step 2.2.1.6.3
Cancel the common factor.
Step 2.2.1.6.4
Rewrite the expression.
Step 2.2.1.7
Multiply by .
Step 2.2.2
Subtract from .
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
Apply the distributive property.
Step 2.3.3
Simplify the expression.
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Step 2.3.3.1
Multiply by .
Step 2.3.3.2
Move to the left of .
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
Move all terms containing to the left side of the equation.
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Step 3.2.1
Add to both sides of the equation.
Step 3.2.2
Add and .
Step 3.3
Add to both sides of the equation.
Step 3.4
Factor using the AC method.
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Step 3.4.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.4.2
Write the factored form using these integers.
Step 3.5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.6
Set equal to and solve for .
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Step 3.6.1
Set equal to .
Step 3.6.2
Subtract from both sides of the equation.
Step 3.7
Set equal to and solve for .
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Step 3.7.1
Set equal to .
Step 3.7.2
Subtract from both sides of the equation.
Step 3.8
The final solution is all the values that make true.