Finite Math Examples

Solve by Substitution x^2+y^2=25 , x^2+4y^2=52
,
Step 1
Solve for in .
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Step 1.1
Subtract from 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
Simplify .
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Step 1.3.1
Rewrite as .
Step 1.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.4.1
First, use the positive value of the to find the first solution.
Step 1.4.2
Next, use the negative value of the to find the second solution.
Step 1.4.3
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
Expand using the FOIL Method.
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Step 2.1.2.1.1.2.1
Apply the distributive property.
Step 2.1.2.1.1.2.2
Apply the distributive property.
Step 2.1.2.1.1.2.3
Apply the distributive property.
Step 2.1.2.1.1.3
Simplify and combine like terms.
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Step 2.1.2.1.1.3.1
Simplify each term.
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Step 2.1.2.1.1.3.1.1
Multiply by .
Step 2.1.2.1.1.3.1.2
Multiply by .
Step 2.1.2.1.1.3.1.3
Move to the left of .
Step 2.1.2.1.1.3.1.4
Rewrite using the commutative property of multiplication.
Step 2.1.2.1.1.3.1.5
Multiply by by adding the exponents.
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Step 2.1.2.1.1.3.1.5.1
Move .
Step 2.1.2.1.1.3.1.5.2
Multiply by .
Step 2.1.2.1.1.3.2
Add and .
Step 2.1.2.1.1.3.3
Add and .
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 2.2.2.3
Simplify the right side.
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Step 2.2.2.3.1
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
Pull terms out from under the radical, assuming positive real numbers.
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 .
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Step 2.3.2.1
Simplify the left side.
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Step 2.3.2.1.1
Remove parentheses.
Step 2.3.2.2
Simplify the right side.
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Step 2.3.2.2.1
Simplify .
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Step 2.3.2.2.1.1
Add and .
Step 2.3.2.2.1.2
Multiply by .
Step 2.3.2.2.1.3
Subtract from .
Step 2.3.2.2.1.4
Multiply by .
Step 2.3.2.2.1.5
Rewrite as .
Step 2.3.2.2.1.6
Pull terms out from under the radical, assuming positive real numbers.
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 .
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Step 2.4.2.1
Simplify the left side.
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Step 2.4.2.1.1
Remove parentheses.
Step 2.4.2.2
Simplify the right side.
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Step 2.4.2.2.1
Simplify .
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Step 2.4.2.2.1.1
Subtract from .
Step 2.4.2.2.1.2
Multiply by .
Step 2.4.2.2.1.3
Add and .
Step 2.4.2.2.1.4
Multiply by .
Step 2.4.2.2.1.5
Rewrite as .
Step 2.4.2.2.1.6
Pull terms out from under the radical, assuming positive real numbers.
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
Expand using the FOIL Method.
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Step 3.1.2.1.1.5.1
Apply the distributive property.
Step 3.1.2.1.1.5.2
Apply the distributive property.
Step 3.1.2.1.1.5.3
Apply the distributive property.
Step 3.1.2.1.1.6
Simplify and combine like terms.
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Step 3.1.2.1.1.6.1
Simplify each term.
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Step 3.1.2.1.1.6.1.1
Multiply by .
Step 3.1.2.1.1.6.1.2
Multiply by .
Step 3.1.2.1.1.6.1.3
Move to the left of .
Step 3.1.2.1.1.6.1.4
Rewrite using the commutative property of multiplication.
Step 3.1.2.1.1.6.1.5
Multiply by by adding the exponents.
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Step 3.1.2.1.1.6.1.5.1
Move .
Step 3.1.2.1.1.6.1.5.2
Multiply by .
Step 3.1.2.1.1.6.2
Add and .
Step 3.1.2.1.1.6.3
Add and .
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 3.2.2.3
Simplify the right side.
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Step 3.2.2.3.1
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
Pull terms out from under the radical, assuming positive real numbers.
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 .
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Step 3.3.2.1
Simplify the left side.
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Step 3.3.2.1.1
Remove parentheses.
Step 3.3.2.2
Simplify the right side.
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Step 3.3.2.2.1
Simplify .
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Step 3.3.2.2.1.1
Add and .
Step 3.3.2.2.1.2
Multiply by .
Step 3.3.2.2.1.3
Subtract from .
Step 3.3.2.2.1.4
Multiply by .
Step 3.3.2.2.1.5
Rewrite as .
Step 3.3.2.2.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 3.3.2.2.1.7
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 .
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Step 3.4.2.1
Simplify the left side.
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Step 3.4.2.1.1
Remove parentheses.
Step 3.4.2.2
Simplify the right side.
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Step 3.4.2.2.1
Simplify .
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Step 3.4.2.2.1.1
Subtract from .
Step 3.4.2.2.1.2
Multiply by .
Step 3.4.2.2.1.3
Add and .
Step 3.4.2.2.1.4
Multiply by .
Step 3.4.2.2.1.5
Rewrite as .
Step 3.4.2.2.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 3.4.2.2.1.7
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