# 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 .

Solve for in the first equation.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Solve for in the second equation.

Simplify.

Simplify .

Simplify each term.

Divide by .

Multiply by .

Add and .

Simplify .

Simplify each term.

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Multiply by .

Add and .

Simplify .

Multiply by .

Add and .

Simplify .

Multiply by .

Divide by .

Add and .

Simplify .

Divide by .

Add and .

Solve the third equation.

Always true

Remove any equations from the system that are always true.

Solve the fourth equation.

Always true

Remove any equations from the system that are always true.

Solve the equation.

Always true

Remove any equations from the system that are always true.

Solve the equation.

Always true

Remove any equations from the system that are always true.

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 .

Calculate the value of when , , and .

Multiply by .

Add and .

Calculate the value of when , , and .

Multiply by .

Add and .

Calculate the value of when , , and .

Multiply by .

Add and .

Calculate the value of when , , and .

Multiply by .

Add and .

Calculate the value of when , , and .

Multiply by .

Add 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.

Use the function rule equation to find .

Simplify.

List all of the solutions.