# Algebra Examples

Step 1

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 .

Step 1.3.1

Solve for in .

Step 1.3.1.1

Rewrite the equation as .

Step 1.3.1.2

Subtract from both sides of the equation.

Step 1.3.2

Replace all occurrences of with in each equation.

Step 1.3.2.1

Replace all occurrences of in with .

Step 1.3.2.2

Simplify the right side.

Step 1.3.2.2.1

Simplify .

Step 1.3.2.2.1.1

Simplify each term.

Step 1.3.2.2.1.1.1

Apply the distributive property.

Step 1.3.2.2.1.1.2

Multiply by .

Step 1.3.2.2.1.1.3

Multiply by .

Step 1.3.2.2.1.2

Add and .

Step 1.3.2.3

Replace all occurrences of in with .

Step 1.3.2.4

Simplify the right side.

Step 1.3.2.4.1

Simplify .

Step 1.3.2.4.1.1

Simplify each term.

Step 1.3.2.4.1.1.1

Apply the distributive property.

Step 1.3.2.4.1.1.2

Multiply by .

Step 1.3.2.4.1.1.3

Multiply by .

Step 1.3.2.4.1.2

Add and .

Step 1.3.2.5

Replace all occurrences of in with .

Step 1.3.2.6

Simplify the right side.

Step 1.3.2.6.1

Simplify .

Step 1.3.2.6.1.1

Simplify each term.

Step 1.3.2.6.1.1.1

Apply the distributive property.

Step 1.3.2.6.1.1.2

Multiply by .

Step 1.3.2.6.1.1.3

Multiply by .

Step 1.3.2.6.1.2

Add and .

Step 1.3.2.7

Replace all occurrences of in with .

Step 1.3.2.8

Simplify the right side.

Step 1.3.2.8.1

Simplify .

Step 1.3.2.8.1.1

Simplify each term.

Step 1.3.2.8.1.1.1

Apply the distributive property.

Step 1.3.2.8.1.1.2

Multiply by .

Step 1.3.2.8.1.1.3

Multiply by .

Step 1.3.2.8.1.2

Add and .

Step 1.3.3

Solve for in .

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.

Step 1.3.3.2.1

Add to both sides of the equation.

Step 1.3.3.2.2

Add and .

Step 1.3.3.3

Divide each term in by and simplify.

Step 1.3.3.3.1

Divide each term in by .

Step 1.3.3.3.2

Simplify the left side.

Step 1.3.3.3.2.1

Cancel the common factor of .

Step 1.3.3.3.2.1.1

Cancel the common factor.

Step 1.3.3.3.2.1.2

Divide by .

Step 1.3.3.3.3

Simplify the right side.

Step 1.3.3.3.3.1

Divide by .

Step 1.3.4

Replace all occurrences of with in each equation.

Step 1.3.4.1

Replace all occurrences of in with .

Step 1.3.4.2

Simplify the right side.

Step 1.3.4.2.1

Simplify .

Step 1.3.4.2.1.1

Multiply by .

Step 1.3.4.2.1.2

Add and .

Step 1.3.4.3

Replace all occurrences of in with .

Step 1.3.4.4

Simplify the right side.

Step 1.3.4.4.1

Simplify .

Step 1.3.4.4.1.1

Multiply by .

Step 1.3.4.4.1.2

Add and .

Step 1.3.4.5

Replace all occurrences of in with .

Step 1.3.4.6

Simplify the right side.

Step 1.3.4.6.1

Simplify .

Step 1.3.4.6.1.1

Multiply by .

Step 1.3.4.6.1.2

Add and .

Step 1.3.4.7

Replace all occurrences of in with .

Step 1.3.4.8

Simplify the right side.

Step 1.3.4.8.1

Simplify .

Step 1.3.4.8.1.1

Multiply by .

Step 1.3.4.8.1.2

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

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 .

Step 2.3.1

Solve for in .

Step 2.3.1.1

Rewrite the equation as .

Step 2.3.1.2

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

Step 2.3.1.2.1

Subtract from both sides of the equation.

Step 2.3.1.2.2

Subtract from both sides of the equation.

Step 2.3.2

Replace all occurrences of with in each equation.

Step 2.3.2.1

Replace all occurrences of in with .

Step 2.3.2.2

Simplify the right side.

Step 2.3.2.2.1

Simplify .

Step 2.3.2.2.1.1

Simplify each term.

Step 2.3.2.2.1.1.1

Raise to the power of .

Step 2.3.2.2.1.1.2

Apply the distributive property.

Step 2.3.2.2.1.1.3

Simplify.

Step 2.3.2.2.1.1.3.1

Multiply by .

Step 2.3.2.2.1.1.3.2

Multiply by .

Step 2.3.2.2.1.1.3.3

Multiply by .

Step 2.3.2.2.1.1.4

Move to the left of .

Step 2.3.2.2.1.2

Simplify by adding terms.

Step 2.3.2.2.1.2.1

Add and .

Step 2.3.2.2.1.2.2

Add and .

Step 2.3.2.3

Replace all occurrences of in with .

Step 2.3.2.4

Simplify the right side.

Step 2.3.2.4.1

Simplify .

Step 2.3.2.4.1.1

Simplify each term.

Step 2.3.2.4.1.1.1

Raise to the power of .

Step 2.3.2.4.1.1.2

Apply the distributive property.

Step 2.3.2.4.1.1.3

Simplify.

Step 2.3.2.4.1.1.3.1

Multiply by .

Step 2.3.2.4.1.1.3.2

Multiply by .

Step 2.3.2.4.1.1.3.3

Multiply by .

Step 2.3.2.4.1.1.4

Move to the left of .

Step 2.3.2.4.1.2

Simplify by adding terms.

Step 2.3.2.4.1.2.1

Add and .

Step 2.3.2.4.1.2.2

Add and .

Step 2.3.2.5

Replace all occurrences of in with .

Step 2.3.2.6

Simplify the right side.

Step 2.3.2.6.1

Simplify .

Step 2.3.2.6.1.1

Simplify each term.

Step 2.3.2.6.1.1.1

Raise to the power of .

Step 2.3.2.6.1.1.2

Apply the distributive property.

Step 2.3.2.6.1.1.3

Simplify.

Step 2.3.2.6.1.1.3.1

Multiply by .

Step 2.3.2.6.1.1.3.2

Multiply by .

Step 2.3.2.6.1.1.3.3

Multiply by .

Step 2.3.2.6.1.1.4

Move to the left of .

Step 2.3.2.6.1.2

Simplify by adding terms.

Step 2.3.2.6.1.2.1

Add and .

Step 2.3.2.6.1.2.2

Add and .

Step 2.3.2.7

Replace all occurrences of in with .

Step 2.3.2.8

Simplify the right side.

Step 2.3.2.8.1

Simplify .

Step 2.3.2.8.1.1

Simplify each term.

Step 2.3.2.8.1.1.1

Raise to the power of .

Step 2.3.2.8.1.1.2

Apply the distributive property.

Step 2.3.2.8.1.1.3

Simplify.

Step 2.3.2.8.1.1.3.1

Multiply by .

Step 2.3.2.8.1.1.3.2

Multiply by .

Step 2.3.2.8.1.1.3.3

Multiply by .

Step 2.3.2.8.1.1.4

Move to the left of .

Step 2.3.2.8.1.2

Simplify by adding terms.

Step 2.3.2.8.1.2.1

Add and .

Step 2.3.2.8.1.2.2

Add and .

Step 2.3.3

Solve for in .

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.

Step 2.3.3.2.1

Add to 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 2.3.3.3

Divide each term in by and simplify.

Step 2.3.3.3.1

Divide each term in by .

Step 2.3.3.3.2

Simplify the left side.

Step 2.3.3.3.2.1

Cancel the common factor of .

Step 2.3.3.3.2.1.1

Cancel the common factor.

Step 2.3.3.3.2.1.2

Divide by .

Step 2.3.3.3.3

Simplify the right side.

Step 2.3.3.3.3.1

Simplify each term.

Step 2.3.3.3.3.1.1

Divide by .

Step 2.3.3.3.3.1.2

Cancel the common factor of and .

Step 2.3.3.3.3.1.2.1

Factor out of .

Step 2.3.3.3.3.1.2.2

Cancel the common factors.

Step 2.3.3.3.3.1.2.2.1

Factor out of .

Step 2.3.3.3.3.1.2.2.2

Cancel the common factor.

Step 2.3.3.3.3.1.2.2.3

Rewrite the expression.

Step 2.3.3.3.3.1.3

Move the negative in front of the fraction.

Step 2.3.4

Replace all occurrences of with in each equation.

Step 2.3.4.1

Replace all occurrences of in with .

Step 2.3.4.2

Simplify the right side.

Step 2.3.4.2.1

Simplify .

Step 2.3.4.2.1.1

Simplify each term.

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 .

Step 2.3.4.2.1.1.3.1

Multiply by .

Step 2.3.4.2.1.1.3.2

Combine and .

Step 2.3.4.2.1.1.3.3

Multiply by .

Step 2.3.4.2.1.2

Add and .

Step 2.3.4.2.1.3

To write as a fraction with a common denominator, multiply by .

Step 2.3.4.2.1.4

Simplify terms.

Step 2.3.4.2.1.4.1

Combine and .

Step 2.3.4.2.1.4.2

Combine the numerators over the common denominator.

Step 2.3.4.2.1.5

Simplify the numerator.

Step 2.3.4.2.1.5.1

Factor out of .

Step 2.3.4.2.1.5.1.1

Factor out of .

Step 2.3.4.2.1.5.1.2

Factor out of .

Step 2.3.4.2.1.5.1.3

Factor out of .

Step 2.3.4.2.1.5.2

Multiply by .

Step 2.3.4.2.1.5.3

Subtract from .

Step 2.3.4.2.1.5.4

Multiply by .

Step 2.3.4.3

Replace all occurrences of in with .

Step 2.3.4.4

Simplify the right side.

Step 2.3.4.4.1

Simplify .

Step 2.3.4.4.1.1

Simplify each term.

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 .

Step 2.3.4.4.1.1.3.1

Multiply by .

Step 2.3.4.4.1.1.3.2

Combine and .

Step 2.3.4.4.1.1.3.3

Multiply by .

Step 2.3.4.4.1.2

Add and .

Step 2.3.4.4.1.3

To write as a fraction with a common denominator, multiply by .

Step 2.3.4.4.1.4

Simplify terms.

Step 2.3.4.4.1.4.1

Combine and .

Step 2.3.4.4.1.4.2

Combine the numerators over the common denominator.

Step 2.3.4.4.1.5

Simplify each term.

Step 2.3.4.4.1.5.1

Simplify the numerator.

Step 2.3.4.4.1.5.1.1

Factor out of .

Step 2.3.4.4.1.5.1.1.1

Factor out of .

Step 2.3.4.4.1.5.1.1.2

Factor out of .

Step 2.3.4.4.1.5.1.1.3

Factor out of .

Step 2.3.4.4.1.5.1.2

Multiply by .

Step 2.3.4.4.1.5.1.3

Subtract from .

Step 2.3.4.4.1.5.2

Move to the left of .

Step 2.3.4.4.1.5.3

Move the negative in front of the fraction.

Step 2.3.4.5

Replace all occurrences of in with .

Step 2.3.4.6

Simplify the right side.

Step 2.3.4.6.1

Simplify .

Step 2.3.4.6.1.1

Simplify each term.

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

Cancel the common factor of .

Step 2.3.4.6.1.1.3.1

Move the leading negative in into the numerator.

Step 2.3.4.6.1.1.3.2

Factor out of .

Step 2.3.4.6.1.1.3.3

Cancel the common factor.

Step 2.3.4.6.1.1.3.4

Rewrite the expression.

Step 2.3.4.6.1.1.4

Multiply by .

Step 2.3.4.6.1.2

Simplify by adding terms.

Step 2.3.4.6.1.2.1

Add and .

Step 2.3.4.6.1.2.2

Subtract from .

Step 2.3.4.7

Replace all occurrences of in with .

Step 2.3.4.8

Simplify the right side.

Step 2.3.4.8.1

Simplify .

Step 2.3.4.8.1.1

Simplify each term.

Step 2.3.4.8.1.1.1

Apply the distributive property.

Step 2.3.4.8.1.1.2

Multiply by .

Step 2.3.4.8.1.1.3

Multiply .

Step 2.3.4.8.1.1.3.1

Multiply by .

Step 2.3.4.8.1.1.3.2

Multiply by .

Step 2.3.4.8.1.2

Add and .

Step 2.3.4.8.1.3

To write as a fraction with a common denominator, multiply by .

Step 2.3.4.8.1.4

Simplify terms.

Step 2.3.4.8.1.4.1

Combine and .

Step 2.3.4.8.1.4.2

Combine the numerators over the common denominator.

Step 2.3.4.8.1.5

Simplify each term.

Step 2.3.4.8.1.5.1

Simplify the numerator.

Step 2.3.4.8.1.5.1.1

Factor out of .

Step 2.3.4.8.1.5.1.1.1

Factor out of .

Step 2.3.4.8.1.5.1.1.2

Factor out of .

Step 2.3.4.8.1.5.1.1.3

Factor out of .

Step 2.3.4.8.1.5.1.2

Multiply by .

Step 2.3.4.8.1.5.1.3

Subtract from .

Step 2.3.4.8.1.5.2

Multiply by .

Step 2.3.5

Solve for in .

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.

Step 2.3.5.2.1

Subtract from both sides of the equation.

Step 2.3.5.2.2

Subtract from .

Step 2.3.5.3

Divide each term in by and simplify.

Step 2.3.5.3.1

Divide each term in by .

Step 2.3.5.3.2

Simplify the left side.

Step 2.3.5.3.2.1

Cancel the common factor of .

Step 2.3.5.3.2.1.1

Cancel the common factor.

Step 2.3.5.3.2.1.2

Divide by .

Step 2.3.5.3.3

Simplify the right side.

Step 2.3.5.3.3.1

Divide by .

Step 2.3.6

Replace all occurrences of with in each equation.

Step 2.3.6.1

Replace all occurrences of in with .

Step 2.3.6.2

Simplify the right side.

Step 2.3.6.2.1

Simplify .

Step 2.3.6.2.1.1

Divide by .

Step 2.3.6.2.1.2

Subtract from .

Step 2.3.6.3

Replace all occurrences of in with .

Step 2.3.6.4

Simplify the right side.

Step 2.3.6.4.1

Simplify .

Step 2.3.6.4.1.1

Simplify each term.

Step 2.3.6.4.1.1.1

Divide by .

Step 2.3.6.4.1.1.2

Multiply by .

Step 2.3.6.4.1.2

Add and .

Step 2.3.6.5

Replace all occurrences of in with .

Step 2.3.6.6

Simplify the right side.

Step 2.3.6.6.1

Simplify .

Step 2.3.6.6.1.1

Multiply by .

Step 2.3.6.6.1.2

Divide by .

Step 2.3.6.6.1.3

Subtract from .

Step 2.3.6.7

Replace all occurrences of in with .

Step 2.3.6.8

Simplify the right side.

Step 2.3.6.8.1

Simplify .

Step 2.3.6.8.1.1

Simplify each term.

Step 2.3.6.8.1.1.1

Cancel the common factor of and .

Step 2.3.6.8.1.1.1.1

Factor out of .

Step 2.3.6.8.1.1.1.2

Cancel the common factors.

Step 2.3.6.8.1.1.1.2.1

Factor out of .

Step 2.3.6.8.1.1.1.2.2

Cancel the common factor.

Step 2.3.6.8.1.1.1.2.3

Rewrite the expression.

Step 2.3.6.8.1.1.1.2.4

Divide by .

Step 2.3.6.8.1.1.2

Multiply .

Step 2.3.6.8.1.1.2.1

Multiply by .

Step 2.3.6.8.1.1.2.2

Multiply by .

Step 2.3.6.8.1.2

Add and .

Step 2.3.7

Remove any equations from the system that are always true.

Step 2.3.8

List all of the solutions.

Step 2.4

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

Step 2.4.1

Calculate the value of such that when , , , and .

Step 2.4.1.1

Simplify each term.

Step 2.4.1.1.1

One to any power is one.

Step 2.4.1.1.2

Multiply by .

Step 2.4.1.1.3

Multiply by .

Step 2.4.1.2

Simplify by adding and subtracting.

Step 2.4.1.2.1

Add and .

Step 2.4.1.2.2

Subtract from .

Step 2.4.2

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

Step 2.4.3

Calculate the value of such that when , , , and .

Step 2.4.3.1

Simplify each term.

Step 2.4.3.1.1

Raise to the power of .

Step 2.4.3.1.2

Multiply by .

Step 2.4.3.1.3

Multiply by .

Step 2.4.3.2

Simplify by adding and subtracting.

Step 2.4.3.2.1

Add and .

Step 2.4.3.2.2

Subtract from .

Step 2.4.4

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

Step 2.4.5

Calculate the value of such that when , , , and .

Step 2.4.5.1

Simplify each term.

Step 2.4.5.1.1

Multiply by by adding the exponents.

Step 2.4.5.1.1.1

Multiply by .

Step 2.4.5.1.1.1.1

Raise to the power of .

Step 2.4.5.1.1.1.2

Use the power rule to combine exponents.

Step 2.4.5.1.1.2

Add and .

Step 2.4.5.1.2

Raise to the power of .

Step 2.4.5.1.3

Multiply by .

Step 2.4.5.2

Simplify by adding and subtracting.

Step 2.4.5.2.1

Add and .

Step 2.4.5.2.2

Subtract from .

Step 2.4.6

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

Step 2.4.7

Calculate the value of such that when , , , and .

Step 2.4.7.1

Simplify each term.

Step 2.4.7.1.1

Raise to the power of .

Step 2.4.7.1.2

Multiply by .

Step 2.4.7.1.3

Multiply by .

Step 2.4.7.2

Simplify by adding and subtracting.

Step 2.4.7.2.1

Add and .

Step 2.4.7.2.2

Subtract from .

Step 2.4.8

Step 2.4.9

Calculate the value of such that when , , , and .

Step 2.4.9.1

Simplify each term.

Step 2.4.9.1.1

Raise to the power of .

Step 2.4.9.1.2

Multiply by .

Step 2.4.9.1.3

Multiply by .

Step 2.4.9.2

Simplify by adding and subtracting.

Step 2.4.9.2.1

Add and .

Step 2.4.9.2.2

Subtract from .

Step 2.4.10

Step 2.4.11

Since for the corresponding values, the function is quadratic.

The function is quadratic

The function is quadratic

The function is quadratic

Step 3

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