Finite Math Examples

Solve by Substitution 3x^2-2y^2+5=0 , 2x^2-y^2+2=0
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Step 1
Solve for in .
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Step 1.1
Move all terms not containing to the right side of the equation.
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Step 1.1.1
Add to both sides of the equation.
Step 1.1.2
Subtract from both sides of the equation.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Move the negative in front of the fraction.
Step 1.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.4
Simplify .
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Step 1.4.1
Combine the numerators over the common denominator.
Step 1.4.2
Rewrite as .
Step 1.4.3
Multiply by .
Step 1.4.4
Combine and simplify the denominator.
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Step 1.4.4.1
Multiply by .
Step 1.4.4.2
Raise to the power of .
Step 1.4.4.3
Raise to the power of .
Step 1.4.4.4
Use the power rule to combine exponents.
Step 1.4.4.5
Add and .
Step 1.4.4.6
Rewrite as .
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Step 1.4.4.6.1
Use to rewrite as .
Step 1.4.4.6.2
Apply the power rule and multiply exponents, .
Step 1.4.4.6.3
Combine and .
Step 1.4.4.6.4
Cancel the common factor of .
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Step 1.4.4.6.4.1
Cancel the common factor.
Step 1.4.4.6.4.2
Rewrite the expression.
Step 1.4.4.6.5
Evaluate the exponent.
Step 1.4.5
Combine using the product rule for radicals.
Step 1.4.6
Reorder factors in .
Step 1.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.5.1
First, use the positive value of the to find the first solution.
Step 1.5.2
Next, use the negative value of the to find the second solution.
Step 1.5.3
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
Apply the product rule to .
Step 2.1.2.1.1.2
Simplify the numerator.
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Step 2.1.2.1.1.2.1
Rewrite as .
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Step 2.1.2.1.1.2.1.1
Use to rewrite as .
Step 2.1.2.1.1.2.1.2
Apply the power rule and multiply exponents, .
Step 2.1.2.1.1.2.1.3
Combine and .
Step 2.1.2.1.1.2.1.4
Cancel the common factor of .
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Step 2.1.2.1.1.2.1.4.1
Cancel the common factor.
Step 2.1.2.1.1.2.1.4.2
Rewrite the expression.
Step 2.1.2.1.1.2.1.5
Simplify.
Step 2.1.2.1.1.2.2
Apply the distributive property.
Step 2.1.2.1.1.2.3
Multiply by .
Step 2.1.2.1.1.2.4
Multiply by .
Step 2.1.2.1.1.2.5
Factor out of .
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Step 2.1.2.1.1.2.5.1
Factor out of .
Step 2.1.2.1.1.2.5.2
Factor out of .
Step 2.1.2.1.1.2.5.3
Factor out of .
Step 2.1.2.1.1.3
Raise to the power of .
Step 2.1.2.1.1.4
Cancel the common factors.
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Step 2.1.2.1.1.4.1
Factor out of .
Step 2.1.2.1.1.4.2
Cancel the common factor.
Step 2.1.2.1.1.4.3
Rewrite the expression.
Step 2.1.2.1.1.5
Combine and .
Step 2.1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.1.2.1.3
Simplify terms.
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Step 2.1.2.1.3.1
Combine and .
Step 2.1.2.1.3.2
Combine the numerators over the common denominator.
Step 2.1.2.1.4
Simplify the numerator.
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Step 2.1.2.1.4.1
Apply the distributive property.
Step 2.1.2.1.4.2
Multiply by .
Step 2.1.2.1.4.3
Multiply by .
Step 2.1.2.1.4.4
Multiply by .
Step 2.1.2.1.4.5
Subtract from .
Step 2.1.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 2.1.2.1.6
Simplify terms.
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Step 2.1.2.1.6.1
Combine and .
Step 2.1.2.1.6.2
Combine the numerators over the common denominator.
Step 2.1.2.1.7
Simplify the numerator.
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Step 2.1.2.1.7.1
Multiply by .
Step 2.1.2.1.7.2
Add and .
Step 2.1.2.1.7.3
Rewrite in a factored form.
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Step 2.1.2.1.7.3.1
Rewrite as .
Step 2.1.2.1.7.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.2
Solve for in .
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Step 2.2.1
Set the numerator equal to zero.
Step 2.2.2
Solve the equation for .
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Step 2.2.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.2.2.2
Set equal to and solve for .
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Step 2.2.2.2.1
Set equal to .
Step 2.2.2.2.2
Subtract from both sides of the equation.
Step 2.2.2.3
Set equal to and solve for .
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Step 2.2.2.3.1
Set equal to .
Step 2.2.2.3.2
Add to both sides of the equation.
Step 2.2.2.4
The final solution is all the values that make true.
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
Simplify the numerator.
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Step 2.3.2.1.1.1
Raise to the power of .
Step 2.3.2.1.1.2
Multiply by .
Step 2.3.2.1.1.3
Subtract from .
Step 2.3.2.1.1.4
Multiply by .
Step 2.3.2.1.1.5
Rewrite as .
Step 2.3.2.1.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 2.3.2.1.2
Divide by .
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
Simplify the numerator.
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Step 2.4.2.1.1.1
Multiply by by adding the exponents.
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Step 2.4.2.1.1.1.1
Multiply by .
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Step 2.4.2.1.1.1.1.1
Raise to the power of .
Step 2.4.2.1.1.1.1.2
Use the power rule to combine exponents.
Step 2.4.2.1.1.1.2
Add and .
Step 2.4.2.1.1.2
Raise to the power of .
Step 2.4.2.1.1.3
Subtract from .
Step 2.4.2.1.1.4
Multiply by .
Step 2.4.2.1.1.5
Rewrite as .
Step 2.4.2.1.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 2.4.2.1.2
Divide by .
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
Use the power rule to distribute the exponent.
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Step 3.1.2.1.1.1.1
Apply the product rule to .
Step 3.1.2.1.1.1.2
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
Simplify the numerator.
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Step 3.1.2.1.1.4.1
Rewrite as .
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Step 3.1.2.1.1.4.1.1
Use to rewrite as .
Step 3.1.2.1.1.4.1.2
Apply the power rule and multiply exponents, .
Step 3.1.2.1.1.4.1.3
Combine and .
Step 3.1.2.1.1.4.1.4
Cancel the common factor of .
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Step 3.1.2.1.1.4.1.4.1
Cancel the common factor.
Step 3.1.2.1.1.4.1.4.2
Rewrite the expression.
Step 3.1.2.1.1.4.1.5
Simplify.
Step 3.1.2.1.1.4.2
Apply the distributive property.
Step 3.1.2.1.1.4.3
Multiply by .
Step 3.1.2.1.1.4.4
Multiply by .
Step 3.1.2.1.1.4.5
Factor out of .
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Step 3.1.2.1.1.4.5.1
Factor out of .
Step 3.1.2.1.1.4.5.2
Factor out of .
Step 3.1.2.1.1.4.5.3
Factor out of .
Step 3.1.2.1.1.5
Raise to the power of .
Step 3.1.2.1.1.6
Cancel the common factors.
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Step 3.1.2.1.1.6.1
Factor out of .
Step 3.1.2.1.1.6.2
Cancel the common factor.
Step 3.1.2.1.1.6.3
Rewrite the expression.
Step 3.1.2.1.1.7
Combine and .
Step 3.1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.1.3
Simplify terms.
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Step 3.1.2.1.3.1
Combine and .
Step 3.1.2.1.3.2
Combine the numerators over the common denominator.
Step 3.1.2.1.4
Simplify the numerator.
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Step 3.1.2.1.4.1
Apply the distributive property.
Step 3.1.2.1.4.2
Multiply by .
Step 3.1.2.1.4.3
Multiply by .
Step 3.1.2.1.4.4
Multiply by .
Step 3.1.2.1.4.5
Subtract from .
Step 3.1.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.1.6
Simplify terms.
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Step 3.1.2.1.6.1
Combine and .
Step 3.1.2.1.6.2
Combine the numerators over the common denominator.
Step 3.1.2.1.7
Simplify the numerator.
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Step 3.1.2.1.7.1
Multiply by .
Step 3.1.2.1.7.2
Add and .
Step 3.1.2.1.7.3
Rewrite in a factored form.
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Step 3.1.2.1.7.3.1
Rewrite as .
Step 3.1.2.1.7.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.2
Solve for in .
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Step 3.2.1
Set the numerator equal to zero.
Step 3.2.2
Solve the equation for .
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Step 3.2.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.2.2.2
Set equal to and solve for .
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Step 3.2.2.2.1
Set equal to .
Step 3.2.2.2.2
Subtract from both sides of the equation.
Step 3.2.2.3
Set equal to and solve for .
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Step 3.2.2.3.1
Set equal to .
Step 3.2.2.3.2
Add to both sides of the equation.
Step 3.2.2.4
The final solution is all the values that make true.
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
Simplify the numerator.
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Step 3.3.2.1.1.1
Raise to the power of .
Step 3.3.2.1.1.2
Multiply by .
Step 3.3.2.1.1.3
Subtract from .
Step 3.3.2.1.1.4
Multiply by .
Step 3.3.2.1.1.5
Rewrite as .
Step 3.3.2.1.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 3.3.2.1.2
Reduce the expression by cancelling the common factors.
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Step 3.3.2.1.2.1
Cancel the common factor of .
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Step 3.3.2.1.2.1.1
Cancel the common factor.
Step 3.3.2.1.2.1.2
Rewrite the expression.
Step 3.3.2.1.2.2
Multiply by .
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
Simplify the numerator.
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Step 3.4.2.1.1.1
Multiply by by adding the exponents.
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Step 3.4.2.1.1.1.1
Multiply by .
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Step 3.4.2.1.1.1.1.1
Raise to the power of .
Step 3.4.2.1.1.1.1.2
Use the power rule to combine exponents.
Step 3.4.2.1.1.1.2
Add and .
Step 3.4.2.1.1.2
Raise to the power of .
Step 3.4.2.1.1.3
Subtract from .
Step 3.4.2.1.1.4
Multiply by .
Step 3.4.2.1.1.5
Rewrite as .
Step 3.4.2.1.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 3.4.2.1.2
Reduce the expression by cancelling the common factors.
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Step 3.4.2.1.2.1
Cancel the common factor of .
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Step 3.4.2.1.2.1.1
Cancel the common factor.
Step 3.4.2.1.2.1.2
Rewrite the expression.
Step 3.4.2.1.2.2
Multiply by .
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