Linear Algebra Examples

Find the Variables [[y,4],[3,2x]]-[[x+3,-3],[2,3y+1]]=[[1,7],[1,-9]]
Step 1
Simplify.
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Step 1.1
Subtract the corresponding elements.
Step 1.2
Simplify each element.
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Step 1.2.1
Simplify each term.
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Step 1.2.1.1
Apply the distributive property.
Step 1.2.1.2
Multiply by .
Step 1.2.2
Add and .
Step 1.2.3
Subtract from .
Step 1.2.4
Simplify each term.
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Step 1.2.4.1
Apply the distributive property.
Step 1.2.4.2
Multiply by .
Step 1.2.4.3
Multiply by .
Step 2
Find the function rule.
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Step 2.1
Check if the function rule is linear.
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Step 2.1.1
To find if the table follows a function rule, check to see if the values follow the linear form .
Step 2.1.2
Build a set of equations from the table such that .
Step 2.1.3
Calculate the values of and .
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Step 2.1.3.1
Solve for in .
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Step 2.1.3.1.1
Rewrite the equation as .
Step 2.1.3.1.2
Subtract from both sides of the equation.
Step 2.1.3.2
Replace all occurrences of with in each equation.
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Step 2.1.3.2.1
Replace all occurrences of in with .
Step 2.1.3.2.2
Simplify the right side.
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Step 2.1.3.2.2.1
Simplify .
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Step 2.1.3.2.2.1.1
Simplify each term.
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Step 2.1.3.2.2.1.1.1
Apply the distributive property.
Step 2.1.3.2.2.1.1.2
Multiply by .
Step 2.1.3.2.2.1.1.3
Multiply by .
Step 2.1.3.2.2.1.2
Add and .
Step 2.1.3.3
Solve for in .
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Step 2.1.3.3.1
Rewrite the equation as .
Step 2.1.3.3.2
Move all terms not containing to the right side of the equation.
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Step 2.1.3.3.2.1
Subtract from both sides of the equation.
Step 2.1.3.3.2.2
Subtract from .
Step 2.1.3.3.3
Divide each term in by and simplify.
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Step 2.1.3.3.3.1
Divide each term in by .
Step 2.1.3.3.3.2
Simplify the left side.
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Step 2.1.3.3.3.2.1
Cancel the common factor of .
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Step 2.1.3.3.3.2.1.1
Cancel the common factor.
Step 2.1.3.3.3.2.1.2
Divide by .
Step 2.1.3.3.3.3
Simplify the right side.
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Step 2.1.3.3.3.3.1
Divide by .
Step 2.1.3.4
Replace all occurrences of with in each equation.
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Step 2.1.3.4.1
Replace all occurrences of in with .
Step 2.1.3.4.2
Simplify the right side.
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Step 2.1.3.4.2.1
Subtract from .
Step 2.1.3.5
List all of the solutions.
Step 2.1.4
Calculate the value of using each value in the relation and compare this value to the given value in the relation.
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Step 2.1.4.1
Calculate the value of when , , and .
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Step 2.1.4.1.1
Multiply by .
Step 2.1.4.1.2
Add and .
Step 2.1.4.2
If the table has a linear function rule, for the corresponding value, . This check passes since and .
Step 2.1.4.3
Calculate the value of when , , and .
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Step 2.1.4.3.1
Multiply by .
Step 2.1.4.3.2
Add and .
Step 2.1.4.4
If the table has a linear function rule, for the corresponding value, . This check passes since and .
Step 2.1.4.5
Since for the corresponding values, the function is linear.
The function is linear
The function is linear
The function is linear
Step 2.2
Since all , the function is linear and follows the form .
Step 3
Find .
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Step 3.1
Use the function rule equation to find .
Step 3.2
Rewrite the equation as .
Step 3.3
Move all terms not containing to the right side of the equation.
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Step 3.3.1
Add to both sides of the equation.
Step 3.3.2
Add to both sides of the equation.
Step 3.3.3
Add and .
Step 4
Find .
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Step 4.1
Use the function rule equation to find .
Step 4.2
Rewrite the equation as .
Step 4.3
Move all terms not containing to the right side of the equation.
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Step 4.3.1
Add to both sides of the equation.
Step 4.3.2
Add to both sides of the equation.
Step 4.3.3
Add and .
Step 4.4
Divide each term in by and simplify.
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Step 4.4.1
Divide each term in by .
Step 4.4.2
Simplify the left side.
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Step 4.4.2.1
Cancel the common factor of .
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Step 4.4.2.1.1
Cancel the common factor.
Step 4.4.2.1.2
Divide by .
Step 4.4.3
Simplify the right side.
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Step 4.4.3.1
Divide by .
Step 5
List all of the solutions.