Algebra Examples

$\begin{array}{c} x & y \\ 1 & 7 \\ 2 & 14 \\ 3 & 21 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.2

Build a set of equations from the table such that $y = a x + b$ .

$7 = a (1) + b 14 = a (2) + b 21 = a (3) + b$

Step 1.3

Calculate the values of $a$ and $b$ .

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Step 1.3.1

Solve for $a$ in $7 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as $a + b = 7$ .

$a + b = 7$

$14 = a (2) + b$

$21 = a (3) + b$

Step 1.3.1.2

Subtract $b$ from both sides of the equation.

$a = 7 - b$

$14 = a (2) + b$

$21 = a (3) + b$

$a = 7 - b$

$14 = a (2) + b$

$21 = a (3) + b$

Step 1.3.2

Replace all occurrences of $a$ with $7 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $a$ in $14 = a (2) + b$ with $7 - b$ .

$14 = (7 - b) (2) + b$

$a = 7 - b$

$21 = a (3) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $(7 - b) (2) + b$ .

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Step 1.3.2.2.1.1

Simplify each term.

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Step 1.3.2.2.1.1.1

Apply the distributive property.

$14 = 7 \cdot 2 - b \cdot 2 + b$

$a = 7 - b$

$21 = a (3) + b$

Step 1.3.2.2.1.1.2

Multiply $7$ by $2$ .

$14 = 14 - b \cdot 2 + b$

$a = 7 - b$

$21 = a (3) + b$

Step 1.3.2.2.1.1.3

Multiply $2$ by $- 1$ .

$14 = 14 - 2 b + b$

$a = 7 - b$

$21 = a (3) + b$

$14 = 14 - 2 b + b$

$a = 7 - b$

$21 = a (3) + b$

Step 1.3.2.2.1.2

Add $- 2 b$ and $b$ .

$14 = 14 - b$

$a = 7 - b$

$21 = a (3) + b$

$14 = 14 - b$

$a = 7 - b$

$21 = a (3) + b$

$14 = 14 - b$

$a = 7 - b$

$21 = a (3) + b$

Step 1.3.2.3

Replace all occurrences of $a$ in $21 = a (3) + b$ with $7 - b$ .

$21 = (7 - b) (3) + b$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify $(7 - b) (3) + b$ .

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Step 1.3.2.4.1.1

Simplify each term.

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Step 1.3.2.4.1.1.1

Apply the distributive property.

$21 = 7 \cdot 3 - b \cdot 3 + b$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.2.4.1.1.2

Multiply $7$ by $3$ .

$21 = 21 - b \cdot 3 + b$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.2.4.1.1.3

Multiply $3$ by $- 1$ .

$21 = 21 - 3 b + b$

$14 = 14 - b$

$a = 7 - b$

$21 = 21 - 3 b + b$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.2.4.1.2

Add $- 3 b$ and $b$ .

$21 = 21 - 2 b$

$14 = 14 - b$

$a = 7 - b$

$21 = 21 - 2 b$

$14 = 14 - b$

$a = 7 - b$

$21 = 21 - 2 b$

$14 = 14 - b$

$a = 7 - b$

$21 = 21 - 2 b$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3

Solve for $b$ in $21 = 21 - 2 b$ .

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Step 1.3.3.1

Rewrite the equation as $21 - 2 b = 21$ .

$21 - 2 b = 21$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $21$ from both sides of the equation.

$- 2 b = 21 - 21$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.2.2

Subtract $21$ from $21$ .

$- 2 b = 0$

$14 = 14 - b$

$a = 7 - b$

$- 2 b = 0$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.3

Divide each term in $- 2 b = 0$ by $- 2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $- 2 b = 0$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{0}{- 2}$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{- 2}$

$14 = 14 - b$

$a = 7 - b$

$b = \frac{0}{- 2}$

$14 = 14 - b$

$a = 7 - b$

$b = \frac{0}{- 2}$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $0$ by $- 2$ .

$b = 0$

$14 = 14 - b$

$a = 7 - b$

$b = 0$

$14 = 14 - b$

$a = 7 - b$

$b = 0$

$14 = 14 - b$

$a = 7 - b$

$b = 0$

$14 = 14 - b$

$a = 7 - b$

Step 1.3.4

Replace all occurrences of $b$ with $0$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $b$ in $14 = 14 - b$ with $0$ .

$14 = 14 - (0)$

$b = 0$

$a = 7 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Subtract $0$ from $14$ .

$14 = 14$

$b = 0$

$a = 7 - b$

$14 = 14$

$b = 0$

$a = 7 - b$

Step 1.3.4.3

Replace all occurrences of $b$ in $a = 7 - b$ with $0$ .

$a = 7 - (0)$

$14 = 14$

$b = 0$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Subtract $0$ from $7$ .

$a = 7$

$14 = 14$

$b = 0$

$a = 7$

$14 = 14$

$b = 0$

$a = 7$

$14 = 14$

$b = 0$

Step 1.3.5

Remove any equations from the system that are always true.

$a = 7$

$b = 0$

Step 1.3.6

List all of the solutions.

$a = 7, b = 0$

Step 1.4

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

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Step 1.4.1

Calculate the value of $y$ when $a = 7$ , $b = 0$ , and $x = 1$ .

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Step 1.4.1.1

Multiply $7$ by $1$ .

$y = 7 + 0$

Step 1.4.1.2

Add $7$ and $0$ .

$y = 7$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 7$ and $y = 7$ .

$7 = 7$

Step 1.4.3

Calculate the value of $y$ when $a = 7$ , $b = 0$ , and $x = 2$ .

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Step 1.4.3.1

Multiply $7$ by $2$ .

$y = 14 + 0$

Step 1.4.3.2

Add $14$ and $0$ .

$y = 14$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 14$ and $y = 14$ .

$14 = 14$

Step 1.4.5

Calculate the value of $y$ when $a = 7$ , $b = 0$ , and $x = 3$ .

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Step 1.4.5.1

Multiply $7$ by $3$ .

$y = 21 + 0$

Step 1.4.5.2

Add $21$ and $0$ .

$y = 21$

Step 1.4.6

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 21$ and $y = 21$ .

$21 = 21$

Step 1.4.7

Since $y = y$ for the corresponding $x$ values, the function is linear.

The function is linear

Step 2

Since all $y = y$ , the function is linear and follows the form $y = 7 x$ .

$y = 7 x$