Finite Math Examples

Solve by Substitution x^2-y^2=21 , 2x^2+y^2=79
,
Step 1
Solve for in .
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Step 1.1
Add to both sides of the equation.
Step 1.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.3.1
First, use the positive value of the to find the first solution.
Step 1.3.2
Next, use the negative value of the to find the second solution.
Step 1.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2
Solve the system .
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Step 2.1
Replace all occurrences of with in each equation.
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Step 2.1.1
Replace all occurrences of in with .
Step 2.1.2
Simplify the left side.
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Step 2.1.2.1
Simplify .
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Step 2.1.2.1.1
Simplify each term.
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Step 2.1.2.1.1.1
Rewrite as .
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Step 2.1.2.1.1.1.1
Use to rewrite as .
Step 2.1.2.1.1.1.2
Apply the power rule and multiply exponents, .
Step 2.1.2.1.1.1.3
Combine and .
Step 2.1.2.1.1.1.4
Cancel the common factor of .
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Step 2.1.2.1.1.1.4.1
Cancel the common factor.
Step 2.1.2.1.1.1.4.2
Rewrite the expression.
Step 2.1.2.1.1.1.5
Simplify.
Step 2.1.2.1.1.2
Apply the distributive property.
Step 2.1.2.1.1.3
Multiply by .
Step 2.1.2.1.2
Add and .
Step 2.2
Solve for in .
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Step 2.2.1
Move all terms not containing to the right side of the equation.
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Step 2.2.1.1
Subtract from both sides of the equation.
Step 2.2.1.2
Subtract from .
Step 2.2.2
Divide each term in by and simplify.
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Step 2.2.2.1
Divide each term in by .
Step 2.2.2.2
Simplify the left side.
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Step 2.2.2.2.1
Cancel the common factor of .
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Step 2.2.2.2.1.1
Cancel the common factor.
Step 2.2.2.2.1.2
Divide by .
Step 2.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.2.4
Simplify .
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Step 2.2.4.1
Rewrite as .
Step 2.2.4.2
Multiply by .
Step 2.2.4.3
Combine and simplify the denominator.
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Step 2.2.4.3.1
Multiply by .
Step 2.2.4.3.2
Raise to the power of .
Step 2.2.4.3.3
Raise to the power of .
Step 2.2.4.3.4
Use the power rule to combine exponents.
Step 2.2.4.3.5
Add and .
Step 2.2.4.3.6
Rewrite as .
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Step 2.2.4.3.6.1
Use to rewrite as .
Step 2.2.4.3.6.2
Apply the power rule and multiply exponents, .
Step 2.2.4.3.6.3
Combine and .
Step 2.2.4.3.6.4
Cancel the common factor of .
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Step 2.2.4.3.6.4.1
Cancel the common factor.
Step 2.2.4.3.6.4.2
Rewrite the expression.
Step 2.2.4.3.6.5
Evaluate the exponent.
Step 2.2.4.4
Simplify the numerator.
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Step 2.2.4.4.1
Combine using the product rule for radicals.
Step 2.2.4.4.2
Multiply by .
Step 2.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 2.2.5.1
First, use the positive value of the to find the first solution.
Step 2.2.5.2
Next, use the negative value of the to find the second solution.
Step 2.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.3
Replace all occurrences of with in each equation.
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Step 2.3.1
Replace all occurrences of in with .
Step 2.3.2
Simplify the right side.
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Step 2.3.2.1
Simplify .
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Step 2.3.2.1.1
Apply the product rule to .
Step 2.3.2.1.2
Rewrite as .
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Step 2.3.2.1.2.1
Use to rewrite as .
Step 2.3.2.1.2.2
Apply the power rule and multiply exponents, .
Step 2.3.2.1.2.3
Combine and .
Step 2.3.2.1.2.4
Cancel the common factor of .
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Step 2.3.2.1.2.4.1
Cancel the common factor.
Step 2.3.2.1.2.4.2
Rewrite the expression.
Step 2.3.2.1.2.5
Evaluate the exponent.
Step 2.3.2.1.3
Raise to the power of .
Step 2.3.2.1.4
Cancel the common factor of and .
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Step 2.3.2.1.4.1
Factor out of .
Step 2.3.2.1.4.2
Cancel the common factors.
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Step 2.3.2.1.4.2.1
Factor out of .
Step 2.3.2.1.4.2.2
Cancel the common factor.
Step 2.3.2.1.4.2.3
Rewrite the expression.
Step 2.3.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 2.3.2.1.6
Combine and .
Step 2.3.2.1.7
Combine the numerators over the common denominator.
Step 2.3.2.1.8
Simplify the numerator.
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Step 2.3.2.1.8.1
Multiply by .
Step 2.3.2.1.8.2
Add and .
Step 2.3.2.1.9
Rewrite as .
Step 2.3.2.1.10
Simplify the numerator.
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Step 2.3.2.1.10.1
Rewrite as .
Step 2.3.2.1.10.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.3.2.1.11
Multiply by .
Step 2.3.2.1.12
Combine and simplify the denominator.
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Step 2.3.2.1.12.1
Multiply by .
Step 2.3.2.1.12.2
Raise to the power of .
Step 2.3.2.1.12.3
Raise to the power of .
Step 2.3.2.1.12.4
Use the power rule to combine exponents.
Step 2.3.2.1.12.5
Add and .
Step 2.3.2.1.12.6
Rewrite as .
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Step 2.3.2.1.12.6.1
Use to rewrite as .
Step 2.3.2.1.12.6.2
Apply the power rule and multiply exponents, .
Step 2.3.2.1.12.6.3
Combine and .
Step 2.3.2.1.12.6.4
Cancel the common factor of .
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Step 2.3.2.1.12.6.4.1
Cancel the common factor.
Step 2.3.2.1.12.6.4.2
Rewrite the expression.
Step 2.3.2.1.12.6.5
Evaluate the exponent.
Step 2.4
Replace all occurrences of with in each equation.
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Step 2.4.1
Replace all occurrences of in with .
Step 2.4.2
Simplify the right side.
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Step 2.4.2.1
Simplify .
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Step 2.4.2.1.1
Use the power rule to distribute the exponent.
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Step 2.4.2.1.1.1
Apply the product rule to .
Step 2.4.2.1.1.2
Apply the product rule to .
Step 2.4.2.1.2
Simplify the expression.
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Step 2.4.2.1.2.1
Raise to the power of .
Step 2.4.2.1.2.2
Multiply by .
Step 2.4.2.1.3
Rewrite as .
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Step 2.4.2.1.3.1
Use to rewrite as .
Step 2.4.2.1.3.2
Apply the power rule and multiply exponents, .
Step 2.4.2.1.3.3
Combine and .
Step 2.4.2.1.3.4
Cancel the common factor of .
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Step 2.4.2.1.3.4.1
Cancel the common factor.
Step 2.4.2.1.3.4.2
Rewrite the expression.
Step 2.4.2.1.3.5
Evaluate the exponent.
Step 2.4.2.1.4
Raise to the power of .
Step 2.4.2.1.5
Cancel the common factor of and .
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Step 2.4.2.1.5.1
Factor out of .
Step 2.4.2.1.5.2
Cancel the common factors.
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Step 2.4.2.1.5.2.1
Factor out of .
Step 2.4.2.1.5.2.2
Cancel the common factor.
Step 2.4.2.1.5.2.3
Rewrite the expression.
Step 2.4.2.1.6
To write as a fraction with a common denominator, multiply by .
Step 2.4.2.1.7
Combine and .
Step 2.4.2.1.8
Combine the numerators over the common denominator.
Step 2.4.2.1.9
Simplify the numerator.
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Step 2.4.2.1.9.1
Multiply by .
Step 2.4.2.1.9.2
Add and .
Step 2.4.2.1.10
Rewrite as .
Step 2.4.2.1.11
Simplify the numerator.
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Step 2.4.2.1.11.1
Rewrite as .
Step 2.4.2.1.11.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.4.2.1.12
Multiply by .
Step 2.4.2.1.13
Combine and simplify the denominator.
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Step 2.4.2.1.13.1
Multiply by .
Step 2.4.2.1.13.2
Raise to the power of .
Step 2.4.2.1.13.3
Raise to the power of .
Step 2.4.2.1.13.4
Use the power rule to combine exponents.
Step 2.4.2.1.13.5
Add and .
Step 2.4.2.1.13.6
Rewrite as .
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Step 2.4.2.1.13.6.1
Use to rewrite as .
Step 2.4.2.1.13.6.2
Apply the power rule and multiply exponents, .
Step 2.4.2.1.13.6.3
Combine and .
Step 2.4.2.1.13.6.4
Cancel the common factor of .
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Step 2.4.2.1.13.6.4.1
Cancel the common factor.
Step 2.4.2.1.13.6.4.2
Rewrite the expression.
Step 2.4.2.1.13.6.5
Evaluate the exponent.
Step 3
Solve the system .
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Step 3.1
Replace all occurrences of with in each equation.
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Step 3.1.1
Replace all occurrences of in with .
Step 3.1.2
Simplify the left side.
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Step 3.1.2.1
Simplify .
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Step 3.1.2.1.1
Simplify each term.
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Step 3.1.2.1.1.1
Apply the product rule to .
Step 3.1.2.1.1.2
Raise to the power of .
Step 3.1.2.1.1.3
Multiply by .
Step 3.1.2.1.1.4
Rewrite as .
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Step 3.1.2.1.1.4.1
Use to rewrite as .
Step 3.1.2.1.1.4.2
Apply the power rule and multiply exponents, .
Step 3.1.2.1.1.4.3
Combine and .
Step 3.1.2.1.1.4.4
Cancel the common factor of .
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Step 3.1.2.1.1.4.4.1
Cancel the common factor.
Step 3.1.2.1.1.4.4.2
Rewrite the expression.
Step 3.1.2.1.1.4.5
Simplify.
Step 3.1.2.1.1.5
Apply the distributive property.
Step 3.1.2.1.1.6
Multiply by .
Step 3.1.2.1.2
Add and .
Step 3.2
Solve for in .
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Step 3.2.1
Move all terms not containing to the right side of the equation.
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Step 3.2.1.1
Subtract from both sides of the equation.
Step 3.2.1.2
Subtract from .
Step 3.2.2
Divide each term in by and simplify.
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Step 3.2.2.1
Divide each term in by .
Step 3.2.2.2
Simplify the left side.
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Step 3.2.2.2.1
Cancel the common factor of .
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Step 3.2.2.2.1.1
Cancel the common factor.
Step 3.2.2.2.1.2
Divide by .
Step 3.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.2.4
Simplify .
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Step 3.2.4.1
Rewrite as .
Step 3.2.4.2
Multiply by .
Step 3.2.4.3
Combine and simplify the denominator.
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Step 3.2.4.3.1
Multiply by .
Step 3.2.4.3.2
Raise to the power of .
Step 3.2.4.3.3
Raise to the power of .
Step 3.2.4.3.4
Use the power rule to combine exponents.
Step 3.2.4.3.5
Add and .
Step 3.2.4.3.6
Rewrite as .
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Step 3.2.4.3.6.1
Use to rewrite as .
Step 3.2.4.3.6.2
Apply the power rule and multiply exponents, .
Step 3.2.4.3.6.3
Combine and .
Step 3.2.4.3.6.4
Cancel the common factor of .
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Step 3.2.4.3.6.4.1
Cancel the common factor.
Step 3.2.4.3.6.4.2
Rewrite the expression.
Step 3.2.4.3.6.5
Evaluate the exponent.
Step 3.2.4.4
Simplify the numerator.
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Step 3.2.4.4.1
Combine using the product rule for radicals.
Step 3.2.4.4.2
Multiply by .
Step 3.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.2.5.1
First, use the positive value of the to find the first solution.
Step 3.2.5.2
Next, use the negative value of the to find the second solution.
Step 3.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.3
Replace all occurrences of with in each equation.
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Step 3.3.1
Replace all occurrences of in with .
Step 3.3.2
Simplify the right side.
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Step 3.3.2.1
Simplify .
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Step 3.3.2.1.1
Apply the product rule to .
Step 3.3.2.1.2
Rewrite as .
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Step 3.3.2.1.2.1
Use to rewrite as .
Step 3.3.2.1.2.2
Apply the power rule and multiply exponents, .
Step 3.3.2.1.2.3
Combine and .
Step 3.3.2.1.2.4
Cancel the common factor of .
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Step 3.3.2.1.2.4.1
Cancel the common factor.
Step 3.3.2.1.2.4.2
Rewrite the expression.
Step 3.3.2.1.2.5
Evaluate the exponent.
Step 3.3.2.1.3
Raise to the power of .
Step 3.3.2.1.4
Cancel the common factor of and .
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Step 3.3.2.1.4.1
Factor out of .
Step 3.3.2.1.4.2
Cancel the common factors.
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Step 3.3.2.1.4.2.1
Factor out of .
Step 3.3.2.1.4.2.2
Cancel the common factor.
Step 3.3.2.1.4.2.3
Rewrite the expression.
Step 3.3.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 3.3.2.1.6
Combine and .
Step 3.3.2.1.7
Combine the numerators over the common denominator.
Step 3.3.2.1.8
Simplify the numerator.
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Step 3.3.2.1.8.1
Multiply by .
Step 3.3.2.1.8.2
Add and .
Step 3.3.2.1.9
Rewrite as .
Step 3.3.2.1.10
Simplify the numerator.
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Step 3.3.2.1.10.1
Rewrite as .
Step 3.3.2.1.10.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.3.2.1.11
Multiply by .
Step 3.3.2.1.12
Combine and simplify the denominator.
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Step 3.3.2.1.12.1
Multiply by .
Step 3.3.2.1.12.2
Raise to the power of .
Step 3.3.2.1.12.3
Raise to the power of .
Step 3.3.2.1.12.4
Use the power rule to combine exponents.
Step 3.3.2.1.12.5
Add and .
Step 3.3.2.1.12.6
Rewrite as .
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Step 3.3.2.1.12.6.1
Use to rewrite as .
Step 3.3.2.1.12.6.2
Apply the power rule and multiply exponents, .
Step 3.3.2.1.12.6.3
Combine and .
Step 3.3.2.1.12.6.4
Cancel the common factor of .
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Step 3.3.2.1.12.6.4.1
Cancel the common factor.
Step 3.3.2.1.12.6.4.2
Rewrite the expression.
Step 3.3.2.1.12.6.5
Evaluate the exponent.
Step 3.4
Replace all occurrences of with in each equation.
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Step 3.4.1
Replace all occurrences of in with .
Step 3.4.2
Simplify the right side.
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Step 3.4.2.1
Simplify .
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Step 3.4.2.1.1
Use the power rule to distribute the exponent.
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Step 3.4.2.1.1.1
Apply the product rule to .
Step 3.4.2.1.1.2
Apply the product rule to .
Step 3.4.2.1.2
Simplify the expression.
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Step 3.4.2.1.2.1
Raise to the power of .
Step 3.4.2.1.2.2
Multiply by .
Step 3.4.2.1.3
Rewrite as .
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Step 3.4.2.1.3.1
Use to rewrite as .
Step 3.4.2.1.3.2
Apply the power rule and multiply exponents, .
Step 3.4.2.1.3.3
Combine and .
Step 3.4.2.1.3.4
Cancel the common factor of .
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Step 3.4.2.1.3.4.1
Cancel the common factor.
Step 3.4.2.1.3.4.2
Rewrite the expression.
Step 3.4.2.1.3.5
Evaluate the exponent.
Step 3.4.2.1.4
Raise to the power of .
Step 3.4.2.1.5
Cancel the common factor of and .
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Step 3.4.2.1.5.1
Factor out of .
Step 3.4.2.1.5.2
Cancel the common factors.
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Step 3.4.2.1.5.2.1
Factor out of .
Step 3.4.2.1.5.2.2
Cancel the common factor.
Step 3.4.2.1.5.2.3
Rewrite the expression.
Step 3.4.2.1.6
To write as a fraction with a common denominator, multiply by .
Step 3.4.2.1.7
Combine and .
Step 3.4.2.1.8
Combine the numerators over the common denominator.
Step 3.4.2.1.9
Simplify the numerator.
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Step 3.4.2.1.9.1
Multiply by .
Step 3.4.2.1.9.2
Add and .
Step 3.4.2.1.10
Rewrite as .
Step 3.4.2.1.11
Simplify the numerator.
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Step 3.4.2.1.11.1
Rewrite as .
Step 3.4.2.1.11.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.4.2.1.12
Multiply by .
Step 3.4.2.1.13
Combine and simplify the denominator.
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Step 3.4.2.1.13.1
Multiply by .
Step 3.4.2.1.13.2
Raise to the power of .
Step 3.4.2.1.13.3
Raise to the power of .
Step 3.4.2.1.13.4
Use the power rule to combine exponents.
Step 3.4.2.1.13.5
Add and .
Step 3.4.2.1.13.6
Rewrite as .
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Step 3.4.2.1.13.6.1
Use to rewrite as .
Step 3.4.2.1.13.6.2
Apply the power rule and multiply exponents, .
Step 3.4.2.1.13.6.3
Combine and .
Step 3.4.2.1.13.6.4
Cancel the common factor of .
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Step 3.4.2.1.13.6.4.1
Cancel the common factor.
Step 3.4.2.1.13.6.4.2
Rewrite the expression.
Step 3.4.2.1.13.6.5
Evaluate the exponent.
Step 4
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 5
The result can be shown in multiple forms.
Point Form:
Equation Form:
Step 6