Question

HHUUURRRYY PLEASE!!!

Complete each table. In the row with x as the input, write a rule as an algebraic expression for the output. Then complete the last row of the table using the rule.

Answer 1

`\begin{array}c\cline{1-2} \sf I\:\!nput & \sf Ou\:\!tput\\\cline{1-2} \sf Muffins & \sf Cost\\\cline{1-2} 1 & 2.25\\\cline{1-2} 3 & 6.75\\\cline{1-2} 6 & 13.50\\\cline{1-2} x & 2.25x\\\cline{1-2} 12 & 27\\\cline{1-2} \end{array}`

Explanation:

Definitions:

• x is the input value.
• y is the output value.

To determine if the relationship is linear, calculate the rate of change between the ordered pairs given in the table:

`\implies \sf rate\;of\;change=(change\:in\:y)/(change\:in\:x)=(13.50-6.75)/(6-3)=(6.75)/(3)=2.25`

`\implies \sf rate\;of\;change=(change\:in\:y)/(change\:in\:x)=(6.75-2.25)/(3-1)=(4.5)/(2)=2.25`

As the rate of change is the same, the relationship is linear.

`\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}`

The rate of change is the slope of a linear equation.

Therefore, to create an equation, substitute the found slope (2.25) and one of the ordered pairs (1, 2.25) into the formula:

`\implies y-2.25=2.25(x-1)`

`\implies y-2.25=2.25x-2.25`

`\implies y=2.25x`

Therefore, the output is 2.25 times the input.

If the input is x, then the output is 2.25x.

If the input is 12, then the output is 2.25 × 12 = 27.

`\begin{array}c\cline{1-2} \sf I\:\!nput & \sf Ou\:\!tput\\\cline{1-2} \sf Muffins & \sf Cost\\\cline{1-2} 1 & 2.25\\\cline{1-2} 3 & 6.75\\\cline{1-2} 6 & 13.50\\\cline{1-2} x & 2.25x\\\cline{1-2} 12 & 27\\\cline{1-2} \end{array}`