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Residents of the town of Maple Grove who are connected to the municipal water supply are billed a fixed amount yearly plus a charge for each cubic foot of water used. A household using \[1000\] cubic feet was billed \[82\], while one using \[1600\] cubic feet was billed \[ 123\].
What is the charge in dollars per cubic foot?

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here we will solve this question by forming two linear equations in the form of \[ax + by = c\] , where\[x\] , and \[y\] are two different variables. Then we will compare the equations to find the answer.

Complete step-by-step answer:
Step 1: It is given in the question that the billed amount equals the fixed amount yearly plus a charge for each cubic foot of water used.
Let, variable \[x\] equals to fixed amount yearly
And the variable \[y\] equals the cubic foot charge of water used. So, the first equation we get:
\[ \Rightarrow x + 1000y = 82\] ……… (1)
Similarly, the second equation will be:
\[ \Rightarrow x + 1600y = 123\] ……… (2)
Step 2: By subtracting equation (1) from equation (2), we get:
\[
  {\text{ }}x + 1600y = 123 \\
   \pm x \pm 1000y = \pm 82 \\
  \;\overline {0x + 600y = 41} \\
 \]
OR \[ \Rightarrow 600y = 41\]
By taking\[600\] into the RHS side, we get:
\[ \Rightarrow y = \dfrac{{41}}{{600}}\]
By dividing the RHS side we get the value of charge in dollars per cubic foot:
\[ \Rightarrow y = \$ 0.06\]

Charge in dollars per cubic foot is \[\$ 0.06\].

Note: We can also solve these types of problems by a graphical method with two different variables \[x\] that denote cubic foot of water used and \[y\]which denotes the total price paid, we have two values of \[x\] i.e.
\[{x_1}\] , \[{x_2}\] and similarly for \[y\] also i.e. \[{y_1}\] , \[{y_2}\].
By using the formula of the slope \[m\] which defines the change in \[y\] over the change in \[x\] of a line:
\[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
By substituting the values of \[{x_1} = 1000\] , \[{x_2} = 1600\], \[{y_1} = 82\] and \[{y_2} = 123\] in the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] , we get:
\[ \Rightarrow m = \dfrac{{123 - 82}}{{1600 - 1000}}\]
By subtracting the numerator and denominator terms in the equation \[m = \dfrac{{123 - 82}}{{1600 - 1000}}\], we get:
\[ \Rightarrow m = \dfrac{{41}}{{600}}\]
By dividing the RHS side we get the value of charge in dollars per cubic foot:
\[ \Rightarrow m = \$ 0.06\]