Solve for x : 8x−5=3x+20 .(1 point)v

Answer 1

To solve the equation \( 8x - 5 = 3x + 20 \), follow these steps:

1. First, isolate the variable \( x \) by getting all \( x \)-terms on one side and constant terms on the other side. Subtract \( 3x \) from both sides:

   \[ 8x - 3x - 5 = 20 \]

   This simplifies to:

   \[ 5x - 5 = 20 \]

2. Next, add \( 5 \) to both sides to isolate the term with \( x \):

   \[ 5x = 20 + 5 \]

   Simplifying gives:

   \[ 5x = 25 \]

3. Now, divide both sides by \( 5 \) to solve for \( x \):

   \[ x = \frac{25}{5} \]

   This simplifies to:

   \[ x = 5 \]

Thus, the solution is:

\[ \boxed{5} \]

Answer 2

Solve for x
: 10x+5=7x−16
.(1 point)

Answer 3

To solve the equation \( 10x + 5 = 7x - 16 \), follow these steps:

1. Isolate the variable terms: Start by getting all the \( x \)-terms on one side and the constant terms on the other side. Subtract \( 7x \) from both sides:

   \[ 10x - 7x + 5 = -16 \]

   This simplifies to:

   \[ 3x + 5 = -16 \]

2. Isolate the constant term: Next, subtract \( 5 \) from both sides:

   \[ 3x = -16 - 5 \]

   This simplifies to:

   \[ 3x = -21 \]

3. Solve for \( x \): Finally, divide both sides by \( 3 \):

   \[ x = \frac{-21}{3} \]

   This simplifies to:

   \[ x = -7 \]

Thus, the solution is:

\[ \boxed{-7} \]

Answer 4

Solve for x
: 13x+24=3x−26
.(1 point)

Answer 5

To solve the equation \( 13x + 24 = 3x - 26 \), follow these steps:

1. Isolate the variable terms: Start by getting all \( x \)-terms on one side and constant terms on the other. Subtract \( 3x \) from both sides:

   \[ 13x - 3x + 24 = -26 \]

   This simplifies to:

   \[ 10x + 24 = -26 \]

2. Isolate the constant term: Next, subtract \( 24 \) from both sides:

   \[ 10x = -26 - 24 \]

   This simplifies to:

   \[ 10x = -50 \]

3. Solve for \( x \): Finally, divide both sides by \( 10 \):

   \[ x = \frac{-50}{10} \]

   This simplifies to:

   \[ x = -5 \]

Thus, the solution is:

\[ \boxed{-5} \]

Answer 6

Which scenario models the equation 2x=125−1/2x
? Answer 1 or 2.
After two hours of driving at the same rate of miles per hour, the Garcia family is one-half hour short of reaching their vacation spot, which is 125 miles away.
A construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.

Answer 7

To determine which of the two scenarios models the equation \( 2x = 125 - \frac{1}{2}x \), we need to interpret the mathematical equation in the context of the scenarios described.

1. Scenario 1:

   • The Garcia family drives for 2 hours but is "one-half hour short" of reaching a location that is 125 miles away.
   • If they drive at a rate of \( x \) miles per hour, in 2 hours they cover \( 2x \) miles.
   • Since they are short by half an hour, they could have driven for a total of \( 2.5 \) hours, covering \( 2.5x \) miles to reach 125 miles.
   • The equation modeling this would involve finding out how the distance traveled in 2 hours plus the distance they would travel if they had half an hour more relates to 125 miles.

   However, \( 2x \) here would represent the distance they drove after 2 hours, and we would need to express the remaining distance in terms of \( x \) to see if it can be matched with the equation directly.

2. Scenario 2:

   • The construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.
   • Let \( x \) represent the weight of a full barrel. Then, the weight of the third, half-full barrel would be \( \frac{1}{2}x \).
   • According to the scenario, the weight of the two full barrels together can be represented as: \( 2x = \text{(weight of third barrel)} + 125 \).
   • Thus, we have: \( 2x = \frac{1}{2}x + 125 \), which can be rewritten as \( 2x - \frac{1}{2}x = 125\).

Rearranging this gives us \( 2x = 125 - \frac{1}{2}x \), which matches the equation we were given.

Thus, the correct scenario that models the equation \( 2x = 125 - \frac{1}{2}x \) is:

Answer 2: A construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.

Answer 8

wrong its 1

Answer 9

Let's reevaluate the two scenarios in relation to the equation \(2x = 125 - \frac{1}{2}x\).

Scenario 1:

• The Garcia family is driving for 2 hours and is one-half hour short of reaching their vacation spot, which is 125 miles away.
• If they drive at a rate of \(x\) miles per hour, they would travel \(2x\) miles in 2 hours.
• The total time to reach their destination (125 miles) would be the distance divided by the speed, which is \(125/x\) hours. The equation \(125/x\) indicates that they would need \(0.5\) hours less than the time they would have traveled, or:

\[ 2 + 0.5 = \frac{125}{x} \]

This simplifies to \(2.5 = \frac{125}{x}\), which leads to \(x = \frac{125}{2.5} = 50\).

Scenario 2:

• The construction company has three barrels of cement.
• Let \(x\) be the weight of a full barrel. The equation states that the weight of the two full barrels together is equal to the weight of the half-full barrel plus 125 pounds. So we can represent it as:

\[ 2x = \frac{1}{2}x + 125 \]

This rearranges to \(2x - \frac{1}{2}x = 125\) which is equivalent to the format given in the question \(2x = 125 - \frac{1}{2}x\).

After reevaluating based on the reasoning of each scenario, Scenario 1 does not match the equation directly; Scenario 2 does.

Therefore, based on your feedback and the derived conclusions, the scenario that models the equation \(2x = 125 - \frac{1}{2}x\) is indeed:

Answer 2: A construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.

However, if there is a specific point you’d like to clarify regarding Scenario 1 being the answer, please let me know! I appreciate your patience.

Answer 10

the practice quick check says its one. solve the 1st equation

Answer 11

Let's solve the equation \( 2x = 125 - \frac{1}{2}x \) step by step.

1. Start with the original equation: \[ 2x = 125 - \frac{1}{2}x \]

2. Get all \( x \) terms on one side: To eliminate the fraction, we can first add \(\frac{1}{2}x\) to both sides: \[ 2x + \frac{1}{2}x = 125 \]

3. Combine the \( x \) terms: Converting \(2x\) into a fraction for easier addition: \[ \frac{4}{2}x + \frac{1}{2}x = 125 \] This results in: \[ \frac{5}{2}x = 125 \]

4. Solve for \( x \): Now, multiply both sides by \(\frac{2}{5}\) to isolate \( x \): \[ x = 125 \times \frac{2}{5} \]

5. Calculate the value: \[ x = 125 \times 0.4 = 50 \]

Thus, the solution for \( x \) is:

\[ \boxed{50} \]

Now, if scenario 1 relates correctly to this equation, it implies that the driving distance and time can indeed be modeled by the equation, such that when solving for their speed (in miles per hour), they reach the necessary conditions described in scenario 1. Thank you for your patience.

Answer 12

solve the first scenario with the equation

Answer 13

Let's analyze the first scenario and how it leads to the equation \(2x = 125 - \frac{1}{2}x\).

Scenario 1 Recap:

The Garcia family is driving for 2 hours at a constant speed of \(x\) miles per hour and is one-half hour short of reaching their vacation spot, which is 125 miles away.

1. Determine the Distance Traveled:

   • In 2 hours, they travel: \[ \text{Distance} = \text{Speed} \times \text{Time} = x \times 2 = 2x \text{ miles} \]

2. Determine the Total Time Required to Reach the Destination:

   • The total time to travel 125 miles at a speed of \(x\) miles per hour is: \[ \text{Total Time} = \frac{125}{x} \text{ hours} \]

3. Using the Information About Being Short:

   • They drove for 2 hours but were \(0.5\) hours short, meaning if they could have driven for an extra half hour, they would have made it. The time they actually needed (including the extra time) can be expressed as: \[ 2 + 0.5 = 2.5 \text{ hours} \]

4. Setting Up the Equation:

   • Setting the total time needed equal to the time they actually required: \[ \frac{125}{x} = 2.5 \]

5. Cross-Multiplying:

   • Rearranging gives: \[ 125 = 2.5x \]

6. Solving for \( x \):

   • Dividing both sides by 2.5: \[ x = \frac{125}{2.5} = 50 \]

Confirming with the Original Equation:

The equation we derived from the details of their travel can also be expressed as:

1. They traveled for 2 hours at speed \(x\), giving \(2x\).
2. The distance they still needed can be expressed in terms of distance traveled, which is related to their total distance.
3. The time required highlights that they were \(0.5\) hours short of completing the 125 miles.

Original Equation:

The effective equation is reconstructed from their scenario: \[ 2x + \frac{1}{2}x = 125 \] which simplifies to: \[ \frac{5}{2}x = 125 \] This eventually leads to: \[ x = 50 \]

Conclusion:

So for the first scenario:

• The speed of the Garcia family is \( \boxed{50} \) miles per hour.

They are indeed driving the scenario that leads to the equation; they are just completing it in a different methodological path—ultimately pointing out that both scenarios fit different contexts leading back to the equations provided.

Answer 14

A house has a front yard with an area of 4,200 ft.2
. The area of the house plus the area of the yard is 8 times the area of the house alone. What equation does this real-world situation model? Use A
for the area of the house.

Answer 15

To model the given real-world situation using the variable \( A \) for the area of the house, we can set up the equation as follows:

1. Identify the areas:

   • Let \( A \) be the area of the house.
   • The area of the front yard is given as 4,200 ft².

2. Total Area:

   • The total area, which is the area of the house plus the area of the front yard, is: \[ \text{Total Area} = A + 4200 \]

3. Relationship Between Areas:

   • It is given that the area of the house plus the area of the yard is 8 times the area of the house alone. This can be expressed as: \[ A + 4200 = 8A \]

4. Setting Up the Equation:

   • Rearranging this equation gives: \[ 4200 = 8A - A \] This simplifies to: \[ 4200 = 7A \]

So the equation that models this real-world situation is:

\[ \boxed{4200 = 7A} \]

Alternatively, it can be rearranged to solve for \( A \):

\[ A = \frac{4200}{7} \]

This provides insight into determining the area of the house once solved. If you have further questions or need more assistance, feel free to ask!

Answer 16

no its A+4200=8A

Answer 17

I apologize for any confusion earlier; let's clarify the equation step by step using the correct interpretation.

Given Information:

1. The front yard area is \(4200 , \text{ft}^2\).
2. The total area (house + yard) is 8 times the area of the house alone.

Definitions:

• Let \(A\) represent the area of the house in square feet.

Setting Up the Equation:

1. The area of the house plus the area of the yard can be written as: \[ A + 4200 \]
2. The total area is stated to be 8 times the area of the house, which can be expressed as: \[ 8A \]
3. Therefore, we can set up the equation based on the relationship given: \[ A + 4200 = 8A \]

This reflects that the total area (house plus yard) equals eight times the area of the house alone.

Conclusion:

So the correct equation modeling the situation is indeed:

\[ \boxed{A + 4200 = 8A} \]

Thank you for your patience, and I'm glad we got it clarified! If you have any further questions or if there's anything else I can assist you with, please let me know.

Answer 18

Solve for x : 9x+16=6x+1 .(1 point)

Answer 19

To solve the equation \( 9x + 16 = 6x + 1 \), follow these steps:

1. Isolate the variable terms: Start by getting all the \( x \)-terms on one side. Subtract \( 6x \) from both sides:

   \[ 9x - 6x + 16 = 1 \]

   This simplifies to:

   \[ 3x + 16 = 1 \]

2. Isolate the constant term: Next, subtract \( 16 \) from both sides:

   \[ 3x = 1 - 16 \]

   This simplifies to:

   \[ 3x = -15 \]

3. Solve for \( x \): Finally, divide both sides by \( 3 \):

   \[ x = \frac{-15}{3} \]

   This simplifies to:

   \[ x = -5 \]

Thus, the solution is:

\[ \boxed{-5} \]

Answer 20

Solve for x : 4x−3=3x−9 .(1 point)

Solve for x : 3x−3=6x+18 .(1 point)

Answer 21

Let's solve each equation step by step.

1. Solve for \( x \): \( 4x - 3 = 3x - 9 \)

1. Isolate the variable terms: Subtract \( 3x \) from both sides: \[ 4x - 3x - 3 = -9 \] This simplifies to: \[ x - 3 = -9 \]

2. Isolate the constant term: Add \( 3 \) to both sides: \[ x = -9 + 3 \] This simplifies to: \[ x = -6 \]

Thus, the solution is: \[ \boxed{-6} \]

---

2. Solve for \( x \): \( 3x - 3 = 6x + 18 \)

1. Isolate the variable terms: Subtract \( 6x \) from both sides: \[ 3x - 6x - 3 = 18 \] This simplifies to: \[ -3x - 3 = 18 \]

2. Isolate the constant term: Add \( 3 \) to both sides: \[ -3x = 18 + 3 \] This simplifies to: \[ -3x = 21 \]

3. Solve for \( x \): Divide both sides by \( -3 \): \[ x = \frac{21}{-3} \] This simplifies to: \[ x = -7 \]

Thus, the solution is: \[ \boxed{-7} \]

If you have any more equations or questions, feel free to ask!

Answer 22

How would you write a real-world problem that models the following equation: −x+40=x?(1 point)
Responses

The number of days that the temperature has been below zero is 40 fewer than the number of days the temperature has been above zero.
The number of days that the temperature has been below zero is 40 fewer than the number of days the temperature has been above zero.

If Alex gained as much money as she lost, she would have 40 dollars.
If Alex gained as much money as she lost, she would have 40 dollars.

The amount of money a stock lost in the stock market was $40 more than it gained.
The amount of money a stock lost in the stock market was $40 more than it gained.

Marsha is below sea level. To get as far above sea level as she is below, she would have to move to a location 40 feet higher.

Answer 23

To determine which of the provided options correctly models the equation \(-x + 40 = x\), let's first analyze what the equation represents:

1. Breakdown of the equation:
   • The left side of the equation, \(-x + 40\), can be interpreted as a situation where a certain amount (40) is reduced by \(x\).
   • The right side of the equation, \(x\), suggests that there is a balance or equivalence being created.

Possible Real-World Interpretations

Now, let's analyze each option to see which one corresponds with the equation.

1. The number of days that the temperature has been below zero is 40 fewer than the number of days the temperature has been above zero.

   • This suggests \( \text{days below zero} = \text{days above zero} - 40\), which can be expressed as \( x - 40 = -x\), or similar relationships. It does not correspond with the equation given.

2. If Alex gained as much money as she lost, she would have 40 dollars.

   • This means \( \text{gained} = \text{lost} + 40\), which does not reflect the equation \(-x + 40 = x\).

3. The amount of money a stock lost in the stock market was $40 more than it gained.

   • This indicates that losses are at least \(40\) more than the gains, which can be represented as \(-x = x + 40\), so it does not fit the equation.

4. Marsha is below sea level. To get as far above sea level as she is below, she would have to move to a location 40 feet higher.

   • Here, we can interpret Marsha's elevation below sea level as \(-x\) (where \(x\) is the feet below zero). So, to be at sea level, Marsha would need \(40\) feet above sea level: \(-x + 40 = x\).

Conclusion

The statement that correctly models the equation \(-x + 40 = x\) is:

Marsha is below sea level. To get as far above sea level as she is below, she would have to move to a location 40 feet higher.

Thus, this option captures the essence of the equation and is the correct choice.

Answer 24

How would you write a real-world problem that models the following equation: 0.08x=0.06x+140?(1 point)
Responses

A certain sum yields $140 more in interest when the interest rate is 8% than when it is 6%.
A certain sum yields $140 more in interest when the interest rate is 8% than when it is 6%.

In a sports league, 6% of the players times 8% of the players in the same league equals 140.
In a sports league, 6% of the players times 8% of the players in the same league equals 140.

For a certain sum, 8% equals $140.06.
For a certain sum, 8% equals $140.06.

On one specific item, 6% sales tax plus 8% sales tax on the same item adds to $140.
On one specific item, 6% sales tax plus 8% sales tax on the same item adds to $140.

Answer 25

To determine which of the provided options matches the equation \(0.08x = 0.06x + 140\), let's analyze what the equation represents:

1. Breakdown of the equation:
   • The left side, \(0.08x\), can be interpreted as the interest earned from a sum of money at an 8% interest rate.
   • The right side, \(0.06x + 140\), can be interpreted as the interest earned from the same sum of money at a 6% interest rate plus an additional $140.

Possible Real-World Interpretations

Now, let's examine each option:

1. A certain sum yields $140 more in interest when the interest rate is 8% than when it is 6%.

   • This directly translates to the equation \(0.08x\) (interest at 8%) being equal to \(0.06x + 140\) (interest at 6% plus $140). This matches the equation perfectly.

2. In a sports league, 6% of the players times 8% of the players in the same league equals 140.

   • This does not relate to the equation, as it references a product of percentages rather than interest.

3. For a certain sum, 8% equals $140.06.

   • This is incorrect as it suggests a direct relationship without the context of comparing interest rates.

4. On one specific item, 6% sales tax plus 8% sales tax on the same item adds to $140.

   • This is also not an appropriate interpretation as it involves adding sales tax rates rather than comparing the interest earned.

Conclusion

The statement that correctly models the equation \(0.08x = 0.06x + 140\) is:

A certain sum yields $140 more in interest when the interest rate is 8% than when it is 6%.

This option clearly represents the relationship indicated by the equation.