# Linear Algebra Examples

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form .

Build a set of equations from the table such that .

Calculate the values of and .

Simplify each equation.

Move to the left of .

Move to the left of .

Move to the left of .

Solve for in the first equation.

Rewrite the equation as .

Subtract from both sides of the equation.

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify each equation.

Remove parentheses.

Subtract from .

Remove parentheses.

Subtract from .

Solve for in the second equation.

Rewrite the equation as .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Replace all occurrences of with in each equation.

Replace all occurrences of in with .

Replace all occurrences of in with .

Simplify.

Simplify .

Multiply by .

Subtract from .

Simplify .

Multiply by .

Add and .

Since , the equation will always be true.

Always true

Remove any equations from the system that are always true.

List the solutions to the system of equations.

Always true

List all of the solutions.

Calculate the value of using each value in the relation and compare this value to the given value in the relation.

Calculate the value of when , , and .

Multiply by .

Add and .

If the table has a linear function rule, for the corresponding value, . This check passes since and .

Calculate the value of when , , and .

Multiply by .

Add and .

If the table has a linear function rule, for the corresponding value, . This check passes since and .

Calculate the value of when , , and .

Multiply by .

Add and .

If the table has a linear function rule, for the corresponding value, . This check passes since and .

Since for the corresponding values, the function is linear.

The function is linear

The function is linear

The function is linear

Since all , the function is linear and follows the form .

Use the function rule equation to find .

Simplify.