Linear Algebra Examples

Find the Domain 9x^2-4y^2=36
Step 1
Subtract from both sides of the equation.
Step 2
Divide each term in by and simplify.
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Step 2.1
Divide 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
Divide by .
Step 2.3
Simplify the right side.
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Step 2.3.1
Simplify each term.
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Step 2.3.1.1
Divide by .
Step 2.3.1.2
Dividing two negative values results in a positive value.
Step 3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4
Simplify .
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Step 4.1
Rewrite as .
Step 4.2
Rewrite as .
Step 4.3
Rewrite as .
Step 4.4
Simplify the expression.
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Step 4.4.1
Rewrite as .
Step 4.4.2
Reorder and .
Step 4.5
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.6
Simplify.
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Step 4.6.1
Factor out of .
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Step 4.6.1.1
Factor out of .
Step 4.6.1.2
Factor out of .
Step 4.6.1.3
Factor out of .
Step 4.6.2
Factor out of .
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Step 4.6.2.1
Factor out of .
Step 4.6.2.2
Factor out of .
Step 4.6.2.3
Factor out of .
Step 4.6.3
Multiply by .
Step 4.7
Write as a fraction with a common denominator.
Step 4.8
Combine the numerators over the common denominator.
Step 4.9
To write as a fraction with a common denominator, multiply by .
Step 4.10
Combine and .
Step 4.11
Combine the numerators over the common denominator.
Step 4.12
Multiply by .
Step 4.13
Combine exponents.
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Step 4.13.1
Combine and .
Step 4.13.2
Multiply by .
Step 4.13.3
Multiply by .
Step 4.14
Rewrite as .
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Step 4.14.1
Factor the perfect power out of .
Step 4.14.2
Factor the perfect power out of .
Step 4.14.3
Rearrange the fraction .
Step 4.15
Pull terms out from under the radical.
Step 4.16
Combine and .
Step 5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 5.1
First, use the positive value of the to find the first solution.
Step 5.2
Next, use the negative value of the to find the second solution.
Step 5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 6
Set the radicand in greater than or equal to to find where the expression is defined.
Step 7
Solve for .
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Step 7.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7.2
Set equal to and solve for .
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Step 7.2.1
Set equal to .
Step 7.2.2
Subtract from both sides of the equation.
Step 7.3
Set equal to and solve for .
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Step 7.3.1
Set equal to .
Step 7.3.2
Add to both sides of the equation.
Step 7.4
The final solution is all the values that make true.
Step 7.5
Use each root to create test intervals.
Step 7.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 7.6.1
Test a value on the interval to see if it makes the inequality true.
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Step 7.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.6.1.2
Replace with in the original inequality.
Step 7.6.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 7.6.2
Test a value on the interval to see if it makes the inequality true.
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Step 7.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.6.2.2
Replace with in the original inequality.
Step 7.6.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 7.6.3
Test a value on the interval to see if it makes the inequality true.
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Step 7.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.6.3.2
Replace with in the original inequality.
Step 7.6.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 7.6.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 7.7
The solution consists of all of the true intervals.
or
or
Step 8
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 9