We are asked to find the rule between the values and fill in the missing entries in the table below.

IN $(x)$	$- 10$	$0$	$5$	$1$	$25$	$- 6$	$8$	$- 1$	$6$	$10$
OUT $(y)$		$5$		$3$	$- 45$	$17$				$- 15$

Rule

Let's start with the rule. We can focus on two inputs,

1

and

25,

because we know their outputs. What we need to do is to evaluate the differences between the inputs and between their corresponding outputs,

3

and

- 45 .

Difference of Inputs : Difference of Outputs : 25 - 1 = 24 - 45 - 3 = - 48

We see that the ratio between the differences equals

- 2 .

\frac{- 4 8}{2 4} = - 2

This suggests that every input is multiplied by

- 2

in the process. Let's check if the inputs multiplied by

- 2

gives the outputs from the table. We will consider only on the inputs that we already have the outputs for.

IN $(x)$	$0$	$1$	$25$	$- 6$	$10$
$- 2 x$	$- 2 (0) = 0$	$- 2 (1) = - 2$	$- 2 (25) = - 50$	$- 2 (- 6) = 12$	$- 2 (10) = - 20$
OUT $(y)$	$5$	$3$	$- 45$	$17$	$- 15$

We can see that in each pair, the input multiplied by $- 2$ is $5$ less than its corresponding output. Let's then also add $5$ and see if this is our rule.

IN $(x)$	$0$	$1$	$25$	$- 6$	$10$
$- 2 x + 5$	$- 2 (0) + 5 = 5$	$- 2 (1) + 5 = 3$	$- 2 (25) + 5 = - 45$	$- 2 (- 6) + 5 = 17$	$- 2 (10) + 5 = - 15$
OUT $(y)$	$5$	$3$	$- 45$	$17$	$- 15$

We can see that the numbers produced and the outputs match now. This means that multiplying the input value by

- 2

and adding

5

gives us the corresponding output. Let's write that algebraically.

- 2 x + 5 = y

Completing the Table

Now we can fill in the gaps in our table. Let's recall the pattern we found.

- 2 x + 5 = y

We can use it to calculate the missing outputs. This will allow us to complete our table!

IN $(x)$	$- 10$	$0$	$5$	$1$	$25$	$- 6$	$8$	$- 1$	$6$	$10$
OUT $(y)$	$- 2 (- 10) + 5 = 25$	$5$	$- 2 (5) + 5 = - 5$	$3$	$- 45$	$17$	$- 2 (8) + 5 = - 11$	$- 2 (- 1) + 5 = 7$	$- 2 (6) + 5 = - 7$	$- 15$

Exercise

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