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A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.)(1 point)

Responses

10−30h=0
10 minus 30 h equals 0

1,280−30h=0
1,280 minus 30 h equals 0

30h=10
30 h equals 10

1,280+30h=0

5 answers

To determine how long the water in the cooler will last, we need an equation that represents the total amount of water (in ounces) and how much is consumed over time.

Given that:

  • There are 1,280 ounces in the cooler (10 gallons = 1,280 ounces).
  • 30 ounces are removed per hour.

Let \( h \) be the number of hours that the cooler has water. The amount of water removed over \( h \) hours is \( 30h \) ounces.

To find when the water will be completely consumed, we can set up the equation:

\[ 1,280 - 30h = 0 \]

This equation shows that when all the water has been removed (the amount left is 0), it equals 1,280 ounces minus how much has been consumed (30 ounces per hour times the number of hours, \( h \)).

So the correct response is:

1,280−30h=0
(1,280 minus 30 h equals 0)

Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x
.(4 points)

sipmle expanitory

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and justify the steps using properties of equality, we'll follow these steps:

Step 1: Distribute

First, we'll distribute the \( 32 \) on the left side of the equation.

\[ 32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x \] This gives us: \[ 128 - 256x \]

Now, substitute back into the equation:

\[ 128 - 256x + 5x = 1 - 7x \]

Step 2: Combine Like Terms

Next, we'll combine the \( x \) terms on the left side:

\[ 128 - 256x + 5x = 128 - 251x \] So the equation becomes:

\[ 128 - 251x = 1 - 7x \]

Step 3: Move All \( x \) Terms to One Side

We'll add \( 251x \) to both sides to isolate the \( x \) terms on the right side:

\[ 128 = 1 - 7x + 251x \] This simplifies to:

\[ 128 = 1 + 244x \]

Step 4: Move Constant Terms

Next, we want to move the constant \( 1 \) to the left by subtracting \( 1 \) from both sides:

\[ 128 - 1 = 244x \]

This results in:

\[ 127 = 244x \]

Step 5: Solve for \( x \)

Now, we divide both sides by \( 244 \):

\[ x = \frac{127}{244} \]

Conclusion: Number of Solutions

The equation \( 32(4 - 8x) + 5x = 1 - 7x \) has a single solution since we were able to isolate \( x \) successfully without encountering contradictory statements or variables that could cancel each other out, indicating that the left and right sides are equal for this value of \( x \).

Therefore, there is exactly one solution to the equation.

in one short thing\

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \):

  1. Distribute: \( 32(4 - 8x) \) becomes \( 128 - 256x \).
  2. Combine like terms: \( 128 - 256x + 5x = 128 - 251x \).
  3. Move \( x \) terms: \( 128 = 1 + 244x \).
  4. Isolate \( x \): \( 127 = 244x \) leads to \( x = \frac{127}{244} \).

Conclusion: The equation has one solution, \( x = \frac{127}{244} \).