# Linear Algebra Examples

Step 1

Step 1.1

Check if the function rule is linear.

Step 1.1.1

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

Step 1.1.2

Build a set of equations from the table such that .

Step 1.1.3

Calculate the values of and .

Step 1.1.3.1

Solve for in .

Step 1.1.3.1.1

Rewrite the equation as .

Step 1.1.3.1.2

Move to the left of .

Step 1.1.3.1.3

Subtract from both sides of the equation.

Step 1.1.3.2

Replace all occurrences of with in each equation.

Step 1.1.3.2.1

Replace all occurrences of in with .

Step 1.1.3.2.2

Simplify .

Step 1.1.3.2.2.1

Simplify the left side.

Step 1.1.3.2.2.1.1

Remove parentheses.

Step 1.1.3.2.2.2

Simplify the right side.

Step 1.1.3.2.2.2.1

Simplify .

Step 1.1.3.2.2.2.1.1

Move to the left of .

Step 1.1.3.2.2.2.1.2

Subtract from .

Step 1.1.3.2.3

Replace all occurrences of in with .

Step 1.1.3.2.4

Simplify .

Step 1.1.3.2.4.1

Simplify the left side.

Step 1.1.3.2.4.1.1

Remove parentheses.

Step 1.1.3.2.4.2

Simplify the right side.

Step 1.1.3.2.4.2.1

Simplify .

Step 1.1.3.2.4.2.1.1

Move to the left of .

Step 1.1.3.2.4.2.1.2

Subtract from .

Step 1.1.3.3

Solve for in .

Step 1.1.3.3.1

Rewrite the equation as .

Step 1.1.3.3.2

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

Step 1.1.3.3.2.1

Subtract from both sides of the equation.

Step 1.1.3.3.2.2

Subtract from .

Step 1.1.3.3.3

Divide each term in by and simplify.

Step 1.1.3.3.3.1

Divide each term in by .

Step 1.1.3.3.3.2

Simplify the left side.

Step 1.1.3.3.3.2.1

Cancel the common factor of .

Step 1.1.3.3.3.2.1.1

Cancel the common factor.

Step 1.1.3.3.3.2.1.2

Divide by .

Step 1.1.3.3.3.3

Simplify the right side.

Step 1.1.3.3.3.3.1

Divide by .

Step 1.1.3.4

Replace all occurrences of with in each equation.

Step 1.1.3.4.1

Replace all occurrences of in with .

Step 1.1.3.4.2

Simplify the right side.

Step 1.1.3.4.2.1

Simplify .

Step 1.1.3.4.2.1.1

Multiply by .

Step 1.1.3.4.2.1.2

Add and .

Step 1.1.3.4.3

Replace all occurrences of in with .

Step 1.1.3.4.4

Simplify the right side.

Step 1.1.3.4.4.1

Simplify .

Step 1.1.3.4.4.1.1

Multiply by .

Step 1.1.3.4.4.1.2

Subtract from .

Step 1.1.3.5

Remove any equations from the system that are always true.

Step 1.1.3.6

List all of the solutions.

Step 1.1.4

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

Step 1.1.4.1

Calculate the value of when , , and .

Step 1.1.4.1.1

Multiply by .

Step 1.1.4.1.2

Add and .

Step 1.1.4.2

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

Step 1.1.4.3

Calculate the value of when , , and .

Step 1.1.4.3.1

Multiply by .

Step 1.1.4.3.2

Add and .

Step 1.1.4.4

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

Step 1.1.4.5

Calculate the value of when , , and .

Step 1.1.4.5.1

Multiply by .

Step 1.1.4.5.2

Add and .

Step 1.1.4.6

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

Step 1.1.4.7

Since for the corresponding values, the function is linear.

The function is linear

The function is linear

The function is linear

Step 1.2

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

Step 2

Step 2.1

Use the function rule equation to find .

Step 2.2

Simplify.