Answer:
[tex]\begin{array}{|c|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}[/tex]
Step-by-step 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:
[tex]\implies \sf rate\;of\;change=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{13.50-6.75}{6-3}=\dfrac{6.75}{3}=2.25[/tex]
[tex]\implies \sf rate\;of\;change=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{6.75-2.25}{3-1}=\dfrac{4.5}{2}=2.25[/tex]
As the rate of change is the same, the relationship is linear.
[tex]\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}}[/tex]
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:
[tex]\implies y-2.25=2.25(x-1)[/tex]
[tex]\implies y-2.25=2.25x-2.25[/tex]
[tex]\implies y=2.25x[/tex]
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.
[tex]\begin{array}{|c|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}[/tex]
