Examples

\begin{matrix} x & q (x) 1 & 1 2 & 2 3 & 3 4 & 4 \end{matrix}

$\begin{array}{c} x & q (x) \\ 1 & 1 \\ 2 & 2 \\ 3 & 3 \\ 4 & 4 \end{array}$

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

$y = a x + b$ .

y = a x + b

$y = a x + b$

Step 1.2

Build a set of equations from the table such that

q (x) = a x + b

$q (x) = a x + b$ .

$1 = a (1) + b 2 = a (2) + b 3 = a (3) + b 4 = a (4) + b$

Step 1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.3.1

Solve for

$a$ in

$1 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as

$a + b = 1$ .

$a + b = 1$

$2 = a (2) + b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.1.2

Subtract

$b$ from both sides of the equation.

$a = 1 - b$

$2 = a (2) + b$

$3 = a (3) + b$

$4 = a (4) + b$

$a = 1 - b$

$2 = a (2) + b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2

Replace all occurrences of

$a$ with

$1 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of

$a$ in

$2 = a (2) + b$ with

$1 - b$ .

$2 = (1 - b) (2) + b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify

$(1 - b) (2) + b$ .

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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.

$2 = 1 \cdot 2 - b \cdot 2 + b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2.2.1.1.2

Multiply

$2$ by

$1$ .

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

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2.2.1.1.3

Multiply

$2$ by

$- 1$ .

$2 = 2 - 2 b + b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

$2 = 2 - 2 b + b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2.2.1.2

Add

$- 2 b$ and

$b$ .

$2 = 2 - b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

$2 = 2 - b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

$2 = 2 - b$

$a = 1 - b$

$3 = a (3) + b$

$4 = a (4) + b$

Step 1.3.2.3

Replace all occurrences of

$a$ in

$3 = a (3) + b$ with

$1 - b$ .

$3 = (1 - b) (3) + b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify

$(1 - b) (3) + b$ .

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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.

$3 = 1 \cdot 3 - b \cdot 3 + b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

Step 1.3.2.4.1.1.2

Multiply

$3$ by

$1$ .

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

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

Step 1.3.2.4.1.1.3

Multiply

$3$ by

$- 1$ .

$3 = 3 - 3 b + b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

$3 = 3 - 3 b + b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

Step 1.3.2.4.1.2

Add

$- 3 b$ and

$b$ .

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = a (4) + b$

Step 1.3.2.5

Replace all occurrences of

$a$ in

$4 = a (4) + b$ with

$1 - b$ .

$4 = (1 - b) (4) + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.2.6

Simplify the right side.

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Step 1.3.2.6.1

Simplify

$(1 - b) (4) + b$ .

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Step 1.3.2.6.1.1

Simplify each term.

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Step 1.3.2.6.1.1.1

Apply the distributive property.

$4 = 1 \cdot 4 - b \cdot 4 + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.2.6.1.1.2

Multiply

$4$ by

$1$ .

$4 = 4 - b \cdot 4 + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.2.6.1.1.3

Multiply

$4$ by

$- 1$ .

$4 = 4 - 4 b + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = 4 - 4 b + b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.2.6.1.2

Add

$- 4 b$ and

$b$ .

$4 = 4 - 3 b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = 4 - 3 b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = 4 - 3 b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$4 = 4 - 3 b$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.3

Solve for

$b$ in

$4 = 4 - 3 b$ .

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Step 1.3.3.1

Rewrite the equation as

$4 - 3 b = 4$ .

$4 - 3 b = 4$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

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

$4$ from both sides of the equation.

$- 3 b = 4 - 4$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.3.2.2

Subtract

$4$ from

$4$ .

$- 3 b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$- 3 b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.3.3

Divide each term in

$- 3 b = 0$ by

$- 3$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$- 3 b = 0$ by

$- 3$ .

$\frac{- 3 b}{- 3} = \frac{0}{- 3}$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

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

$- 3$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 b}{- 3} = \frac{0}{- 3}$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.3.3.2.1.2

Divide

$b$ by

$1$ .

$b = \frac{0}{- 3}$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$b = \frac{0}{- 3}$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$b = \frac{0}{- 3}$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$0$ by

$- 3$ .

$b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

$b = 0$

$3 = 3 - 2 b$

$2 = 2 - b$

$a = 1 - b$

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

$3 = 3 - 2 b$ with

$0$ .

$3 = 3 - 2 \cdot 0$

$b = 0$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify

$3 - 2 \cdot 0$ .

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Step 1.3.4.2.1.1

Multiply

$- 2$ by

$0$ .

$3 = 3 + 0$

$b = 0$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.4.2.1.2

Add

$3$ and

$0$ .

$3 = 3$

$b = 0$

$2 = 2 - b$

$a = 1 - b$

$3 = 3$

$b = 0$

$2 = 2 - b$

$a = 1 - b$

$3 = 3$

$b = 0$

$2 = 2 - b$

$a = 1 - b$

Step 1.3.4.3

Replace all occurrences of

$b$ in

$2 = 2 - b$ with

$0$ .

$2 = 2 - (0)$

$3 = 3$

$b = 0$

$a = 1 - b$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Subtract

$0$ from

$2$ .

$2 = 2$

$3 = 3$

$b = 0$

$a = 1 - b$

$2 = 2$

$3 = 3$

$b = 0$

$a = 1 - b$

Step 1.3.4.5

Replace all occurrences of

$b$ in

$a = 1 - b$ with

$0$ .

$a = 1 - (0)$

$2 = 2$

$3 = 3$

$b = 0$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Subtract

$0$ from

$1$ .

$a = 1$

$2 = 2$

$3 = 3$

$b = 0$

$a = 1$

$2 = 2$

$3 = 3$

$b = 0$

$a = 1$

$2 = 2$

$3 = 3$

$b = 0$

Step 1.3.5

Remove any equations from the system that are always true.

$a = 1$

$b = 0$

Step 1.3.6

List all of the solutions.

$a = 1, b = 0$

Step 1.4

Calculate the value of

$y$ using each

$x$ value in the relation and compare this value to the given

$q (x)$ value in the relation.

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Step 1.4.1

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = 1$ .

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Step 1.4.1.1

Multiply

$1$ by

$1$ .

$y = 1 + 0$

Step 1.4.1.2

Add

$1$ and

$0$ .

$y = 1$

Step 1.4.2

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 1$ . This check passes since

$y = 1$ and

$q (x) = 1$ .

$1 = 1$

Step 1.4.3

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = 2$ .

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Step 1.4.3.1

Multiply

$2$ by

$1$ .

$y = 2 + 0$

Step 1.4.3.2

Add

$2$ and

$0$ .

$y = 2$

Step 1.4.4

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 2$ . This check passes since

$y = 2$ and

$q (x) = 2$ .

$2 = 2$

Step 1.4.5

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = 3$ .

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Step 1.4.5.1

Multiply

$3$ by

$1$ .

$y = 3 + 0$

Step 1.4.5.2

Add

$3$ and

$0$ .

$y = 3$

Step 1.4.6

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 3$ . This check passes since

$y = 3$ and

$q (x) = 3$ .

$3 = 3$

Step 1.4.7

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = 4$ .

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Step 1.4.7.1

Multiply

$4$ by

$1$ .

$y = 4 + 0$

Step 1.4.7.2

Add

$4$ and

$0$ .

$y = 4$

Step 1.4.8

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 4$ . This check passes since

$y = 4$ and

$q (x) = 4$ .

$4 = 4$

Step 1.4.9

Since

$y = q (x)$ for the corresponding

$x$ values, the function is linear.

The function is linear

Step 2

Since all

$y = q (x)$ , the function is linear and follows the form

$y = x$ .

$y = x$

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