an office building has three air - conditioning units, one for each office. each unit runs only when…

an office building has three air - conditioning units, one for each office. each unit runs only when employees are working. the tables below show the number of hours employees worked at each office over a three - month period and the total air conditioning electric costs for the entire office for each month.\n\n| | unit a | unit b | unit c |\n|--|--|--|--|\n| june | 184 | 207 | 276 |\n| july | 174 | 198 | 264 |\n| august | 168 | 189 | 242 |\n\n| | total air conditioning electric costs |\n|--|--|\n| june | $142.60 |\n| july | $135.96 |\n| august | $128.30 |\n\nhow much does it cost to run unit a per hour?

an office building has three air - conditioning units, one for each office. each unit runs only when employees are working. the tables below show the number of hours employees worked at each office over a three - month period and the total air conditioning electric costs for the entire office for each month.\n\n| | unit a | unit b | unit c |\n|--|--|--|--|\n| june | 184 | 207 | 276 |\n| july | 174 | 198 | 264 |\n| august | 168 | 189 | 242 |\n\n| | total air conditioning electric costs |\n|--|--|\n| june | $142.60 |\n| july | $135.96 |\n| august | $128.30 |\n\nhow much does it cost to run unit a per hour?

Answer

Explanation:

Step1: Define variables and formulate equations.

Let $x$, $y$, and $z$ be the hourly running costs for Unit A, Unit B, and Unit C, respectively. Based on the provided data, we can set up a system of linear equations: For June: $184x + 207y + 276z = 142.60$ (1) For July: $174x + 198y + 264z = 135.96$ (2) For August: $168x + 189y + 242z = 128.30$ (3)

Step2: Eliminate one variable to create a system of two equations.

Subtract Equation (2) from Equation (1): $(184x - 174x) + (207y - 198y) + (276z - 264z) = 142.60 - 135.96$ $10x + 9y + 12z = 6.64$ (4) Subtract Equation (3) from Equation (2): $(174x - 168x) + (198y - 189y) + (264z - 242z) = 135.96 - 128.30$ $6x + 9y + 22z = 7.66$ (5)

Step3: Eliminate another variable from the new system.

Subtract Equation (4) from Equation (5) to eliminate $y$: $(6x - 10x) + (9y - 9y) + (22z - 12z) = 7.66 - 6.64$ $-4x + 10z = 1.02$ (6)

Step4: Express $z$ in terms of $x$.

From Equation (6): $10z = 1.02 + 4x$ $z = \frac{1.02 + 4x}{10} = 0.102 + 0.4x$ (7)

Step5: Express $y$ in terms of $x$.

Substitute $z$ from Equation (7) into Equation (4): $10x + 9y + 12(0.102 + 0.4x) = 6.64$ $10x + 9y + 1.224 + 4.8x = 6.64$ $14.8x + 9y + 1.224 = 6.64$ $9y = 6.64 - 1.224 - 14.8x$ $9y = 5.416 - 14.8x$ $y = \frac{5.416 - 14.8x}{9}$ (8)

Step6: Substitute expressions for $y$ and $z$ into an original equation.

Substitute $y$ from Equation (8) and $z$ from Equation (7) into Equation (3): $168x + 189\left(\frac{5.416 - 14.8x}{9}\right) + 242(0.102 + 0.4x) = 128.30$ $168x + 21(5.416 - 14.8x) + 24.684 + 96.8x = 128.30$ $168x + 113.736 - 310.8x + 24.684 + 96.8x = 128.30$

Step7: Solve for $x$.

Combine terms involving $x$: $168x - 310.8x + 96.8x = (168 + 96.8 - 310.8)x = (264.8 - 310.8)x = -46x$ Combine constant terms: $113.736 + 24.684 = 138.42$ The equation becomes: $-46x + 138.42 = 128.30$ $-46x = 128.30 - 138.42$ $-46x = -10.12$ $x = \frac{-10.12}{-46}$ $x = 0.22$

Answer:

$0.22