Linear Algebra Examples

Find the Variables [[2,7],[5,3]]x=[[-3,8],[7,-9]]
Step 1
Check if the function rule is linear.
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Step 1.1
To find if the table follows a function rule, check to see if the values follow the linear form .
Step 1.2
Build a set of equations from the table such that .
Step 1.3
Calculate the values of and .
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Step 1.3.1
Solve for in .
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Step 1.3.1.1
Rewrite the equation as .
Step 1.3.1.2
Move to the left of .
Step 1.3.1.3
Subtract from both sides of the equation.
Step 1.3.2
Replace all occurrences of with in each equation.
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Step 1.3.2.1
Replace all occurrences of in with .
Step 1.3.2.2
Simplify .
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Step 1.3.2.2.1
Simplify the left side.
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Step 1.3.2.2.1.1
Remove parentheses.
Step 1.3.2.2.2
Simplify the right side.
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Step 1.3.2.2.2.1
Simplify .
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Step 1.3.2.2.2.1.1
Move to the left of .
Step 1.3.2.2.2.1.2
Subtract from .
Step 1.3.2.3
Replace all occurrences of in with .
Step 1.3.2.4
Simplify .
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Step 1.3.2.4.1
Simplify the left side.
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Step 1.3.2.4.1.1
Remove parentheses.
Step 1.3.2.4.2
Simplify the right side.
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Step 1.3.2.4.2.1
Simplify .
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Step 1.3.2.4.2.1.1
Move to the left of .
Step 1.3.2.4.2.1.2
Subtract from .
Step 1.3.2.5
Replace all occurrences of in with .
Step 1.3.2.6
Simplify .
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Step 1.3.2.6.1
Simplify the left side.
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Step 1.3.2.6.1.1
Remove parentheses.
Step 1.3.2.6.2
Simplify the right side.
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Step 1.3.2.6.2.1
Simplify .
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Step 1.3.2.6.2.1.1
Move to the left of .
Step 1.3.2.6.2.1.2
Subtract from .
Step 1.3.3
Solve for in .
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Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Move all terms not containing to the right side of the equation.
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Step 1.3.3.2.1
Add to both sides of the equation.
Step 1.3.3.2.2
Add and .
Step 1.3.4
Replace all occurrences of with in each equation.
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Step 1.3.4.1
Replace all occurrences of in with .
Step 1.3.4.2
Simplify the right side.
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Step 1.3.4.2.1
Simplify .
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Step 1.3.4.2.1.1
Multiply by .
Step 1.3.4.2.1.2
Subtract from .
Step 1.3.4.3
Replace all occurrences of in with .
Step 1.3.4.4
Simplify the right side.
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Step 1.3.4.4.1
Simplify .
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Step 1.3.4.4.1.1
Multiply by .
Step 1.3.4.4.1.2
Subtract from .
Step 1.3.4.5
Replace all occurrences of in with .
Step 1.3.4.6
Simplify the right side.
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Step 1.3.4.6.1
Simplify .
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Step 1.3.4.6.1.1
Multiply by .
Step 1.3.4.6.1.2
Add and .
Step 1.3.5
Since is not true, there is no solution.
No solution
No solution
Step 1.4
Since for the corresponding values, the function is not linear.
The function is not linear
The function is not linear
Step 2
Check if the function rule is quadratic.
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Step 2.1
To find if the table follows a function rule, check whether the function rule could follow the form .
Step 2.2
Build a set of equations from the table such that .
Step 2.3
Calculate the values of , , and .
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Step 2.3.1
Solve for in .
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Step 2.3.1.1
Rewrite the equation as .
Step 2.3.1.2
Simplify each term.
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Step 2.3.1.2.1
Raise to the power of .
Step 2.3.1.2.2
Move to the left of .
Step 2.3.1.2.3
Move to the left of .
Step 2.3.1.3
Move all terms not containing to the right side of the equation.
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Step 2.3.1.3.1
Subtract from both sides of the equation.
Step 2.3.1.3.2
Subtract from both sides of the equation.
Step 2.3.2
Replace all occurrences of with in each equation.
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Step 2.3.2.1
Replace all occurrences of in with .
Step 2.3.2.2
Simplify .
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Step 2.3.2.2.1
Simplify the left side.
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Step 2.3.2.2.1.1
Remove parentheses.
Step 2.3.2.2.2
Simplify the right side.
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Step 2.3.2.2.2.1
Simplify .
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Step 2.3.2.2.2.1.1
Simplify each term.
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Step 2.3.2.2.2.1.1.1
Raise to the power of .
Step 2.3.2.2.2.1.1.2
Move to the left of .
Step 2.3.2.2.2.1.1.3
Move to the left of .
Step 2.3.2.2.2.1.2
Simplify by adding terms.
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Step 2.3.2.2.2.1.2.1
Subtract from .
Step 2.3.2.2.2.1.2.2
Subtract from .
Step 2.3.2.3
Replace all occurrences of in with .
Step 2.3.2.4
Simplify .
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Step 2.3.2.4.1
Simplify the left side.
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Step 2.3.2.4.1.1
Remove parentheses.
Step 2.3.2.4.2
Simplify the right side.
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Step 2.3.2.4.2.1
Simplify .
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Step 2.3.2.4.2.1.1
Simplify each term.
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Step 2.3.2.4.2.1.1.1
Raise to the power of .
Step 2.3.2.4.2.1.1.2
Move to the left of .
Step 2.3.2.4.2.1.1.3
Move to the left of .
Step 2.3.2.4.2.1.2
Simplify by adding terms.
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Step 2.3.2.4.2.1.2.1
Subtract from .
Step 2.3.2.4.2.1.2.2
Subtract from .
Step 2.3.2.5
Replace all occurrences of in with .
Step 2.3.2.6
Simplify .
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Step 2.3.2.6.1
Simplify the left side.
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Step 2.3.2.6.1.1
Remove parentheses.
Step 2.3.2.6.2
Simplify the right side.
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Step 2.3.2.6.2.1
Simplify .
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Step 2.3.2.6.2.1.1
Simplify each term.
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Step 2.3.2.6.2.1.1.1
Raise to the power of .
Step 2.3.2.6.2.1.1.2
Move to the left of .
Step 2.3.2.6.2.1.1.3
Move to the left of .
Step 2.3.2.6.2.1.2
Simplify by adding terms.
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Step 2.3.2.6.2.1.2.1
Subtract from .
Step 2.3.2.6.2.1.2.2
Subtract from .
Step 2.3.3
Solve for in .
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Step 2.3.3.1
Rewrite the equation as .
Step 2.3.3.2
Move all terms not containing to the right side of the equation.
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Step 2.3.3.2.1
Subtract from both sides of the equation.
Step 2.3.3.2.2
Add to both sides of the equation.
Step 2.3.3.2.3
Add and .
Step 2.3.4
Replace all occurrences of with in each equation.
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Step 2.3.4.1
Replace all occurrences of in with .
Step 2.3.4.2
Simplify the right side.
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Step 2.3.4.2.1
Simplify .
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Step 2.3.4.2.1.1
Simplify each term.
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Step 2.3.4.2.1.1.1
Apply the distributive property.
Step 2.3.4.2.1.1.2
Multiply by .
Step 2.3.4.2.1.1.3
Multiply by .
Step 2.3.4.2.1.2
Simplify by adding terms.
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Step 2.3.4.2.1.2.1
Subtract from .
Step 2.3.4.2.1.2.2
Subtract from .
Step 2.3.4.3
Replace all occurrences of in with .
Step 2.3.4.4
Simplify the right side.
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Step 2.3.4.4.1
Simplify .
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Step 2.3.4.4.1.1
Simplify each term.
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Step 2.3.4.4.1.1.1
Apply the distributive property.
Step 2.3.4.4.1.1.2
Multiply by .
Step 2.3.4.4.1.1.3
Multiply by .
Step 2.3.4.4.1.2
Simplify by adding terms.
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Step 2.3.4.4.1.2.1
Subtract from .
Step 2.3.4.4.1.2.2
Subtract from .
Step 2.3.4.5
Replace all occurrences of in with .
Step 2.3.4.6
Simplify the right side.
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Step 2.3.4.6.1
Simplify .
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Step 2.3.4.6.1.1
Simplify each term.
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Step 2.3.4.6.1.1.1
Apply the distributive property.
Step 2.3.4.6.1.1.2
Multiply by .
Step 2.3.4.6.1.1.3
Multiply by .
Step 2.3.4.6.1.2
Simplify by adding terms.
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Step 2.3.4.6.1.2.1
Add and .
Step 2.3.4.6.1.2.2
Add and .
Step 2.3.5
Solve for in .
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Step 2.3.5.1
Rewrite the equation as .
Step 2.3.5.2
Move all terms not containing to the right side of the equation.
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Step 2.3.5.2.1
Add to both sides of the equation.
Step 2.3.5.2.2
Add and .
Step 2.3.5.3
Divide each term in by and simplify.
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Step 2.3.5.3.1
Divide each term in by .
Step 2.3.5.3.2
Simplify the left side.
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Step 2.3.5.3.2.1
Cancel the common factor of .
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Step 2.3.5.3.2.1.1
Cancel the common factor.
Step 2.3.5.3.2.1.2
Divide by .
Step 2.3.6
Replace all occurrences of with in each equation.
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Step 2.3.6.1
Replace all occurrences of in with .
Step 2.3.6.2
Simplify the right side.
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Step 2.3.6.2.1
Simplify .
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Step 2.3.6.2.1.1
Simplify each term.
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Step 2.3.6.2.1.1.1
Cancel the common factor of .
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Step 2.3.6.2.1.1.1.1
Factor out of .
Step 2.3.6.2.1.1.1.2
Factor out of .
Step 2.3.6.2.1.1.1.3
Cancel the common factor.
Step 2.3.6.2.1.1.1.4
Rewrite the expression.
Step 2.3.6.2.1.1.2
Combine and .
Step 2.3.6.2.1.1.3
Multiply by .
Step 2.3.6.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.6.2.1.3
Combine and .
Step 2.3.6.2.1.4
Combine the numerators over the common denominator.
Step 2.3.6.2.1.5
Simplify the numerator.
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Step 2.3.6.2.1.5.1
Multiply by .
Step 2.3.6.2.1.5.2
Add and .
Step 2.3.6.3
Replace all occurrences of in with .
Step 2.3.6.4
Simplify the right side.
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Step 2.3.6.4.1
Simplify .
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Step 2.3.6.4.1.1
Simplify each term.
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Step 2.3.6.4.1.1.1
Cancel the common factor of .
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Step 2.3.6.4.1.1.1.1
Factor out of .
Step 2.3.6.4.1.1.1.2
Factor out of .
Step 2.3.6.4.1.1.1.3
Cancel the common factor.
Step 2.3.6.4.1.1.1.4
Rewrite the expression.
Step 2.3.6.4.1.1.2
Combine and .
Step 2.3.6.4.1.1.3
Multiply by .
Step 2.3.6.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.6.4.1.3
Combine and .
Step 2.3.6.4.1.4
Combine the numerators over the common denominator.
Step 2.3.6.4.1.5
Simplify the numerator.
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Step 2.3.6.4.1.5.1
Multiply by .
Step 2.3.6.4.1.5.2
Subtract from .
Step 2.3.6.4.1.6
Move the negative in front of the fraction.
Step 2.3.6.5
Replace all occurrences of in with .
Step 2.3.6.6
Simplify the right side.
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Step 2.3.6.6.1
Simplify .
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Step 2.3.6.6.1.1
Simplify each term.
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Step 2.3.6.6.1.1.1
Cancel the common factor of .
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Step 2.3.6.6.1.1.1.1
Factor out of .
Step 2.3.6.6.1.1.1.2
Factor out of .
Step 2.3.6.6.1.1.1.3
Cancel the common factor.
Step 2.3.6.6.1.1.1.4
Rewrite the expression.
Step 2.3.6.6.1.1.2
Rewrite as .
Step 2.3.6.6.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.6.6.1.3
Combine and .
Step 2.3.6.6.1.4
Combine the numerators over the common denominator.
Step 2.3.6.6.1.5
Simplify the numerator.
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Step 2.3.6.6.1.5.1
Multiply by .
Step 2.3.6.6.1.5.2
Subtract from .
Step 2.3.6.6.1.6
Move the negative in front of the fraction.
Step 2.3.7
Since is not true, there is no solution.
No solution
No solution
Step 2.4
Calculate the value of using each value in the table and compare this value to the given value in the table.
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Step 2.4.1
Calculate the value of such that when , , , and .
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Step 2.4.1.1
Simplify each term.
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Step 2.4.1.1.1
Raise to the power of .
Step 2.4.1.1.2
Multiply by .
Step 2.4.1.1.3
Multiply by .
Step 2.4.1.2
Simplify by adding numbers.
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Step 2.4.1.2.1
Add and .
Step 2.4.1.2.2
Add and .
Step 2.4.2
If the table has a quadratic function rule, for the corresponding value, . This check does not pass, since and . The function rule can't be quadratic.
Step 2.4.3
Since for the corresponding values, the function is not quadratic.
The function is not quadratic
The function is not quadratic
The function is not quadratic
Step 3
Check if the function rule is cubic.
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Step 3.1
To find if the table follows a function rule, check whether the function rule could follow the form .
Step 3.2
Build a set of equations from the table such that .
Step 3.3
Calculate the values of , , , and .
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Step 3.3.1
Solve for in .
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Step 3.3.1.1
Rewrite the equation as .
Step 3.3.1.2
Simplify each term.
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Step 3.3.1.2.1
Raise to the power of .
Step 3.3.1.2.2
Move to the left of .
Step 3.3.1.2.3
Raise to the power of .
Step 3.3.1.2.4
Move to the left of .
Step 3.3.1.2.5
Move to the left of .
Step 3.3.1.3
Move all terms not containing to the right side of the equation.
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Step 3.3.1.3.1
Subtract from both sides of the equation.
Step 3.3.1.3.2
Subtract from both sides of the equation.
Step 3.3.1.3.3
Subtract from both sides of the equation.
Step 3.3.2
Replace all occurrences of with in each equation.
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Step 3.3.2.1
Replace all occurrences of in with .
Step 3.3.2.2
Simplify .
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Step 3.3.2.2.1
Simplify the left side.
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Step 3.3.2.2.1.1
Remove parentheses.
Step 3.3.2.2.2
Simplify the right side.
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Step 3.3.2.2.2.1
Simplify .
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Step 3.3.2.2.2.1.1
Combine the opposite terms in .
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Step 3.3.2.2.2.1.1.1
Reorder the factors in the terms and .
Step 3.3.2.2.2.1.1.2
Subtract from .
Step 3.3.2.2.2.1.1.3
Add and .
Step 3.3.2.2.2.1.2
Simplify each term.
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Step 3.3.2.2.2.1.2.1
Raise to the power of .
Step 3.3.2.2.2.1.2.2
Move to the left of .
Step 3.3.2.2.2.1.2.3
Raise to the power of .
Step 3.3.2.2.2.1.2.4
Move to the left of .
Step 3.3.2.2.2.1.3
Simplify by adding terms.
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Step 3.3.2.2.2.1.3.1
Combine the opposite terms in .
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Step 3.3.2.2.2.1.3.1.1
Subtract from .
Step 3.3.2.2.2.1.3.1.2
Add and .
Step 3.3.2.2.2.1.3.2
Subtract from .
Step 3.3.2.3
Replace all occurrences of in with .
Step 3.3.2.4
Simplify .
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Step 3.3.2.4.1
Simplify the left side.
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Step 3.3.2.4.1.1
Remove parentheses.
Step 3.3.2.4.2
Simplify the right side.
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Step 3.3.2.4.2.1
Simplify .
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Step 3.3.2.4.2.1.1
Combine the opposite terms in .
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Step 3.3.2.4.2.1.1.1
Reorder the factors in the terms and .
Step 3.3.2.4.2.1.1.2
Subtract from .
Step 3.3.2.4.2.1.1.3
Add and .
Step 3.3.2.4.2.1.2
Simplify each term.
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Step 3.3.2.4.2.1.2.1
Raise to the power of .
Step 3.3.2.4.2.1.2.2
Move to the left of .
Step 3.3.2.4.2.1.2.3
Raise to the power of .
Step 3.3.2.4.2.1.2.4
Move to the left of .
Step 3.3.2.4.2.1.3
Simplify by adding terms.
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Step 3.3.2.4.2.1.3.1
Combine the opposite terms in .
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Step 3.3.2.4.2.1.3.1.1
Subtract from .
Step 3.3.2.4.2.1.3.1.2
Add and .
Step 3.3.2.4.2.1.3.2
Subtract from .
Step 3.3.2.5
Replace all occurrences of in with .
Step 3.3.2.6
Simplify .
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Step 3.3.2.6.1
Simplify the left side.
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Step 3.3.2.6.1.1
Remove parentheses.
Step 3.3.2.6.2
Simplify the right side.
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Step 3.3.2.6.2.1
Simplify .
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Step 3.3.2.6.2.1.1
Combine the opposite terms in .
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Step 3.3.2.6.2.1.1.1
Reorder the factors in the terms and .
Step 3.3.2.6.2.1.1.2
Subtract from .
Step 3.3.2.6.2.1.1.3
Add and .
Step 3.3.2.6.2.1.2
Simplify each term.
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Step 3.3.2.6.2.1.2.1
Raise to the power of .
Step 3.3.2.6.2.1.2.2
Move to the left of .
Step 3.3.2.6.2.1.2.3
Raise to the power of .
Step 3.3.2.6.2.1.2.4
Move to the left of .
Step 3.3.2.6.2.1.3
Simplify by adding terms.
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Step 3.3.2.6.2.1.3.1
Combine the opposite terms in .
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Step 3.3.2.6.2.1.3.1.1
Subtract from .
Step 3.3.2.6.2.1.3.1.2
Add and .
Step 3.3.2.6.2.1.3.2
Subtract from .
Step 3.3.3
Solve for in .
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Step 3.3.3.1
Rewrite the equation as .
Step 3.3.3.2
Move all terms not containing to the right side of the equation.
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Step 3.3.3.2.1
Add to both sides of the equation.
Step 3.3.3.2.2
Add and .
Step 3.3.3.3
Divide each term in by and simplify.
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Step 3.3.3.3.1
Divide each term in by .
Step 3.3.3.3.2
Simplify the left side.
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Step 3.3.3.3.2.1
Cancel the common factor of .
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Step 3.3.3.3.2.1.1
Cancel the common factor.
Step 3.3.3.3.2.1.2
Divide by .
Step 3.3.3.3.3
Simplify the right side.
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Step 3.3.3.3.3.1
Move the negative in front of the fraction.
Step 3.3.4
Replace all occurrences of with in each equation.
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Step 3.3.4.1
Replace all occurrences of in with .
Step 3.3.4.2
Simplify the right side.
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Step 3.3.4.2.1
Simplify .
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Step 3.3.4.2.1.1
Simplify each term.
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Step 3.3.4.2.1.1.1
Multiply .
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Step 3.3.4.2.1.1.1.1
Multiply by .
Step 3.3.4.2.1.1.1.2
Combine and .
Step 3.3.4.2.1.1.1.3
Multiply by .
Step 3.3.4.2.1.1.2
Move the negative in front of the fraction.
Step 3.3.4.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.3.4.2.1.3
Combine and .
Step 3.3.4.2.1.4
Combine the numerators over the common denominator.
Step 3.3.4.2.1.5
Simplify the numerator.
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Step 3.3.4.2.1.5.1
Multiply by .
Step 3.3.4.2.1.5.2
Subtract from .
Step 3.3.4.2.1.6
Move the negative in front of the fraction.
Step 3.3.4.3
Replace all occurrences of in with .
Step 3.3.4.4
Simplify the right side.
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Step 3.3.4.4.1
Simplify .
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Step 3.3.4.4.1.1
Simplify each term.
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Step 3.3.4.4.1.1.1
Multiply .
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Step 3.3.4.4.1.1.1.1
Multiply by .
Step 3.3.4.4.1.1.1.2
Combine and .
Step 3.3.4.4.1.1.1.3
Multiply by .
Step 3.3.4.4.1.1.2
Move the negative in front of the fraction.
Step 3.3.4.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.3.4.4.1.3
Combine and .
Step 3.3.4.4.1.4
Combine the numerators over the common denominator.
Step 3.3.4.4.1.5
Simplify the numerator.
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Step 3.3.4.4.1.5.1
Multiply by .
Step 3.3.4.4.1.5.2
Subtract from .
Step 3.3.4.4.1.6
Move the negative in front of the fraction.
Step 3.3.4.5
Replace all occurrences of in with .
Step 3.3.4.6
Simplify the right side.
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Step 3.3.4.6.1
Simplify .
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Step 3.3.4.6.1.1
Multiply .
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Step 3.3.4.6.1.1.1
Multiply by .
Step 3.3.4.6.1.1.2
Combine and .
Step 3.3.4.6.1.1.3
Multiply by .
Step 3.3.4.6.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.3.4.6.1.3
Combine and .
Step 3.3.4.6.1.4
Combine the numerators over the common denominator.
Step 3.3.4.6.1.5
Simplify the numerator.
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Step 3.3.4.6.1.5.1
Multiply by .
Step 3.3.4.6.1.5.2
Add and .
Step 3.3.4.6.1.6
Move the negative in front of the fraction.
Step 3.3.5
Since is not true, there is no solution.
No solution
No solution
Step 3.4
Calculate the value of using each value in the table and compare this value to the given value in the table.
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Step 3.4.1
Calculate the value of such that when , , , , and .
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Step 3.4.1.1
Simplify each term.
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Step 3.4.1.1.1
Raise to the power of .
Step 3.4.1.1.2
Multiply by .
Step 3.4.1.1.3
Raise to the power of .
Step 3.4.1.1.4
Multiply by .
Step 3.4.1.1.5
Multiply .
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Step 3.4.1.1.5.1
Multiply by .
Step 3.4.1.1.5.2
Multiply by .
Step 3.4.1.2
Simplify by adding numbers.
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Step 3.4.1.2.1
Add and .
Step 3.4.1.2.2
Add and .
Step 3.4.2
If the table has a cubic function rule, for the corresponding value, . This check does not pass, since and . The function rule can't be cubic.
Step 3.4.3
Since for the corresponding values, the function is not cubic.
The function is not cubic
The function is not cubic
The function is not cubic
Step 4
There are no values for , , , and in the equations , , and that work for every pair of and .
The table does not have a function rule that is linear, quadratic, or cubic.