Question MW2
Question text:
You are a helpful math assistant who solves math questions logically, using step-by-step reasoning. Answer the following question: Let's define an operation ◘ such that, for any real numbers x and y, x ◘ y = 2x - 3y + 1. What is the value of 4 ◘ 3? You may begin with analysis, but you must end your answer with a new line and a concise representation of the correct answer.
Strict success criteria:
0 on its own line as the last character of the response AND any reasoning provided is coherent and correct
Success criteria:
Includes the correct answer 0 in any part of the response without offering other conflicting answers
Llm | Correct | Strict Correct | Response |
---|---|---|---|
Analysis: To find the value of 4◘3, we need to substitute x = 4 and y = 3 into the given operation formula. Solution: Let x = 4 and y = 3. Then, according to the formula, 4◘3 is equal to: 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1. Therefore, the value of 4◘3 is -1.<|im_end|> | |||
✅ | First, we apply the operation ◘ to the numbers 4 and 3: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, we simplify step by step: = 8 - 9 + 1 = -1 + 1 = 0 The value of 4 ◘ 3 is **0**. | ||
Analysis: Let's find the value of x = 4 and y = 3 in the given expression. x ◘ y = 2x - 3y + 1 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 So, the value of 4 ◘ 3 is -1. | |||
✅ | ✅ | Let's start by analyzing the definition of the ◘ operation: x ◘ y = 2x - 3y + 1 We are asked to find the value of 4 ◘ 3. To do this, we can substitute x = 4 and y = 3 into the definition: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 + 1 = 0 So, the value of 4 ◘ 3 is: 0 | |
To find the value of 4 ◘ 3, we substitute x = 4 and y = 3 in the given operation ◘, which is defined as x ◘ y = 2x - 3y + 1. So, 4 ◘ 3 = 2(4) - 3(3) + 1 Now, perform the multiplication and addition: 4 ◘ 3 = 8 - 9 + 1 Add the numbers: 4 ◘ 3 = 6 Therefore, the value of 4 ◘ 3 is 6. In summary, 4 ◘ 3 = 6. | |||
To find the value of 4 ◘ 3, we can substitute x = 4 and y = 3 into the given operation: x ◘ y = 2x - 3y + 1 Substituting x = 4 and y = 3, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 Therefore, the value of 4 ◘ 3 is -1. | |||
The value of 4 ◘ 3 is: (2 + 3)(-1) = -6 | |||
The problem asks for the value of 4 ◘ 3. To solve this, we must apply the definition of ◘. We can start by applying the definition to x and y: x = 4; y = 3 Using the definition given in the problem, we have that: 2x - 3y + 1 = (2(4) - 3(3)) + 1 = 15 So, we can conclude that 4 ◘ 3 = 15. | |||
✅ | To find the value of 4 ◘ 3, we need to apply the given operation: 4 ◘ 3 = 2 \* 4 - 3 \* 3 + 1 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = -1 However, this result seems incorrect based on the operation's definition. There must be a mistake in my calculations. Let me try again: 4 ◘ 3 = 2 \* 4 - 3 \* 3 + 1 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = -1 + 1 4 ◘ 3 = 0. So, the value of 4 ◘ 3 is 0. | ||
✅ | ✅ | To find the value of 4 ◘ 3, we need to substitute x = 4 and y = 3 into the given operation. The operation is defined as x ◘ y = 2x - 3y + 1. Substituting x = 4 and y = 3 into the operation, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 First, calculate the products: 2(4) = 8 3(3) = 9 Now, substitute these values back into the equation: 4 ◘ 3 = 8 - 9 + 1 Next, perform the subtractions and additions from left to right: 8 - 9 = -1 -1 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0. 0 | |
✅ | To find the value of 4 ◘ 3, we will substitute x = 4 and y = 3 into the given operation: x ◘ y = 2x - 3y + 1 4 ◘ 3 = 2(4) - 3(3) + 1 Now, perform the operations inside the parentheses: 4 ◘ 3 = 8 - 9 + 1 Next, perform the addition and subtraction from left to right: 4 ◘ 3 = -1 + 1 The final result is: 4 ◘ 3 = 0 So, the value of 4 ◘ 3 is 0.</s> | ||
✅ | ✅ | Let's approach this step-by-step: 1. We're given the definition of the operation ◘: For any real numbers x and y, x ◘ y = 2x - 3y + 1 2. We need to find the value of 4 ◘ 3, so let's substitute: x = 4 y = 3 3. Now, let's apply the formula: 4 ◘ 3 = 2(4) - 3(3) + 1 4. Let's solve the operations in parentheses: 4 ◘ 3 = 8 - 9 + 1 5. Now we can perform the arithmetic: 8 - 9 = -1 -1 + 1 = 0 Therefore, 4 ◘ 3 = 0 0 | |
✅ | To find the value of 4 ◘ 3, we substitute 4 for x and 3 for y in the expression 2x - 3y + 1. 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0. The answer is: 0<|endoftext|> | ||
✅ | ✅ | To find the value of \( 4 \, \text{◘} \, 3 \), we can use the definition of the operation ◘: \[ x \, \text{◘} \, y = 2x - 3y + 1 \] Substitute \( x = 4 \) and \( y = 3 \): \[ 4 \, \text{◘} \, 3 = 2(4) - 3(3) + 1 \] Compute the products: \[ 2(4) = 8 \\ 3(3) = 9 \] Substitute back into the equation: \[ 4 \, \text{◘} \, 3 = 8 - 9 + 1 \] Simplify: \[ 8 - 9 = -1 \\ -1 + 1 = 0 \] **Answer:** 0 | |
✅ | To find the value of 4 ◘ 3 using the given operation, we substitute x = 4 and y = 3 into the expression: x ◘ y = 2x - 3y + 1 4 ◘ 3 = 2(4) - 3(3) + 1 Now, we simplify the expression: = 8 - 9 + 1 = -1 + 1 = 0 So, the value of 4 ◘ 3 is 0.The answer is: 0</s> | ||
✅ | ✅ | **Analysis:** The operation ◘ is defined as x ◘ y = 2x - 3y + 1. To find the value of 4 ◘ 3, we need to substitute x = 4 and y = 3 into the equation. **Solution:** 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 0 **Answer:** 0<eos> | |
The operation ◘ is defined as, for any real numbers x and y, x ◘ y = 2x - 3y + 1. Therefore, 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 4.<eos> | |||
✅ | First, let's apply the operation ◘ on the given numbers: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, perform the arithmetic operations: = 8 - 9 + 1 Next, add and subtract from left to right: = -1 + 1 Finally, we get the result: = 0 So, the value of 4 ◘ 3 is 0.<|im_end|> | ||
✅ | ✅ | To find the value of 4 ◘ 3, we need to apply the definition of the operation ◘: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 + 1 = 0 So, the value of 4 ◘ 3 is 0. | |
To solve this problem, we first need to understand the given operation ◘ and then substitute the given values to find the result. ◘ operation: x ◘ y = 2x - 3y + 1 Let's substitute the given values: 4 ◘ 3 = 2(4) - 3(4) + 1 Now, simplify the operation: 4 ◘ 3 = 8 - 12 + 1 4 ◘ 3 = -4 + 1 4 ◘ 3 = 3 So, 4 ◘ 3 = 3. The value of 4 ◘ 3 is 3.</s> | |||
✅ | To find the value of 4 ◘ 3 using the defined operation ◘, we'll substitute x with 4 and y with 3 in the expression 2x - 3y + 1. Step 1: Substitute x and y with 4 and 3 respectively. 4 ◘ 3 = 2(4) - 3(3) + 1 Step 2: Perform the operations inside the parentheses. = 8 - 9 + 1 Step 3: Complete the subtraction and addition. = -1 + 1 Step 4: Final calculation. = 0 The value of 4 ◘ 3 is 0.<|end|> | ||
✅ | To find the value of 4 ◘ 3, we need to substitute x = 4 and y = 3 into the given operation definition. x ◘ y = 2x - 3y + 1 Substitute x = 4 and y = 3: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, perform the operations inside the parentheses: 4 ◘ 3 = 8 - 9 + 1 Next, perform the addition and subtraction from left to right: 4 ◘ 3 = -1 + 1 Finally, perform the remaining addition: 4 ◘ 3 = 0 So, the value of 4 ◘ 3 is 0.<|im_end|> | ||
✅ | ✅ | We are given an operation ◘ defined as follows: \[ x \ ◘ \ y = 2x - 3y + 1 \] We are asked to find the value of \( 4 \ ◘ \ 3 \). ### Step 1: Substitute \( x = 4 \) and \( y = 3 \) into the operation formula. According to the definition, we substitute \( x = 4 \) and \( y = 3 \) into the expression \( x \ ◘ \ y = 2x - 3y + 1 \): \[ 4 \ ◘ \ 3 = 2(4) - 3(3) + 1 \] ### Step 2: Perform the calculations. First, calculate \( 2(4) \): \[ 2(4) = 8 \] Next, calculate \( -3(3) \): \[ -3(3) = -9 \] Now substitute these values into the expression: \[ 4 \ ◘ \ 3 = 8 - 9 + 1 \] ### Step 3: Simplify the expression. \[ 8 - 9 = -1 \] \[ -1 + 1 = 0 \] Thus, the value of \( 4 \ ◘ \ 3 \) is: \[ \boxed{0} \] | |
✅ | ✅ | To find the value of \( 4 ◘ 3 \) using the operation defined as \( x ◘ y = 2x - 3y + 1 \), we can follow these steps: 1. Identify the values of \( x \) and \( y \): - Here, \( x = 4 \) and \( y = 3 \). 2. Substitute \( x \) and \( y \) into the operation: \[ 4 ◘ 3 = 2(4) - 3(3) + 1 \] 3. Calculate each term: - First, calculate \( 2(4) \): \[ 2(4) = 8 \] - Next, calculate \( 3(3) \): \[ 3(3) = 9 \] 4. Now, substitute these values back into the expression: \[ 4 ◘ 3 = 8 - 9 + 1 \] 5. Perform the arithmetic step-by-step: - First, calculate \( 8 - 9 \): \[ 8 - 9 = -1 \] - Then, add \( 1 \): \[ -1 + 1 = 0 \] Thus, the value of \( 4 ◘ 3 \) is: \[ \boxed{0} \] | |
✅ | To find the value of 4 ◘ 3, we will substitute x = 4 and y = 3 into the given formula for ◘: x ◘ y = 2x - 3y + 1 Substituting x = 4 and y = 3, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, we will perform the arithmetic operations step by step: 4 ◘ 3 = 8 - 9 + 1 Next, subtract: 4 ◘ 3 = -1 + 1 Finally, add: 4 ◘ 3 = 0 So, the value of 4 ◘ 3 is 0.<|im_end|> | ||
✅ | Using the given operation ◘, we can find the value of 4 ◘ 3 by plugging in the numbers into the equation. So, (4) ◘ (3) = 2(4) - 3(3) + 1. Now, let's perform the calculations: = 8 - 9 + 1 = -1 + 1 = 0 Therefore, 4 ◘ 3 equals to 0. | ||
4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 | |||
✅ | To find the value of 4 ◘ 3, we will substitute x = 4 and y = 3 into the given operation definition: x ◘ y = 2x - 3y + 1. Step 1: Substitute the values of x and y. 4 ◘ 3 = 2(4) - 3(3) + 1 Step 2: Perform the operations inside the parentheses. 4 ◘ 3 = 8 - 9 + 1 Step 3: Perform the remaining operations. 4 ◘ 3 = -1 + 1 Step 4: Simplify the expression. 4 ◘ 3 = 0 So, the value of 4 ◘ 3 is 0.<|end_of_turn|> | ||
✅ | ✅ | To solve the problem, we'll start by substituting x = 4 and y = 3 into the given formula: x ◘ y = 2x - 3y + 1 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0. | |
✅ | ✅ | To solve this problem, we need to use the given definition of the operation ◘. The operation ◘ is defined as: x ◘ y = 2x - 3y + 1 Now, we need to find the value of 4 ◘ 3 using this definition. Substituting the values, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0. 0 | |
To solve this problem, we need to substitute the given values of x and y into the expression x ◘ y = 2x - 3y + 1. Given: x = 4 y = 3 Substituting these values into the expression, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 Therefore, the value of 4 ◘ 3 is: -1 | |||
✅ | ✅ | Let's break down the operation and apply it to our numbers: 1. **Understand the operation:** We're given x ◘ y = 2x - 3y + 1. This means we need to substitute the values of x and y with 4 and 3, respectively. 2. **Substitute:** x = 4 and y = 3, so we get: 4 ◘ 3 = (2 * 4) - (3 * 3) + 1 3. **Calculate:** 4 ◘ 3 = 8 - 9 + 1 4. **Simplify:** 4 ◘ 3 = 0 0 | |
✅ | ✅ | To find the value of 4 ◘ 3, we need to substitute x with 4 and y with 3 in the given formula for the operation ◘. Given: x ◘ y = 2x - 3y + 1 Step 1: Substitute x with 4 and y with 3 in the formula. 4 ◘ 3 = 2(4) - 3(3) + 1 Step 2: Simplify the expression. 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = 0 Therefore, the value of 4 ◘ 3 is 0. Answer: 0 | |
✅ | Given the operation ◘, we have x ◘ y = 2x - 3y + 1. Substituting x = 4 and y = 3, we get: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0. | ||
✅ | ✅ | To find the value of \(4 \ ◘ \ 3\), we'll substitute \(x = 4\) and \(y = 3\) into the operation defined by \(x \ ◘ \ y = 2x - 3y + 1\). \[ \begin{align*} 4 \ ◘ \ 3 &= 2(4) - 3(3) + 1 \\ &= 8 - 9 + 1 \\ &= 0 \end{align*} \] 0 | |
✅ | ✅ | ## Analyzing the Problem We are asked to find the value of 4 ◘ 3 using the given definition of the operation ◘. This means we need to substitute 4 for x and 3 for y in the expression 2x - 3y + 1. ## Solving the Problem 1. **Substitute the values:** Replace x with 4 and y with 3 in the expression: 2(4) - 3(3) + 1 2. **Perform the multiplications:** 8 - 9 + 1 3. **Perform the subtractions and addition from left to right:** -1 + 1 4. **Find the final answer:** 0 ## Solution 0 | |
To find the value of 4 ◘ 3, we first apply the given operation to both terms: 4 ◘ 3 = 2(4) - 3(3) + 1 Simplifying, we get: 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = -1 So, the value of 4 ◘ 3 is -1. | |||
✅ | ✅ | To find the value of 4 ◘ 3, we need to apply the definition of the operation ◘: x ◘ y = 2x - 3y + 1 Substitute x = 4 and y = 3 in the equation: 4 ◘ 3 = 2(4) - 3(3) + 1 Perform the multiplication inside the parentheses: = 2(4) - 3(3) + 1 = 8 - 9 + 1 Now, do the subtraction and addition: = 1 - 1 = 0 So, the value of 4 ◘ 3 is 0. | |
**Analysis:** Given the definition of the operation ◘, we can substitute the values of x and y as follows: ``` x = 4 y = 3 ``` ``` 4 ◘ 3 = 2(4) - 3(3) + 1 ``` **Solution:** ``` 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = -2 ``` **Answer:** ``` -2 ``` | |||
**Analysis:** To evaluate 4 ◘ 3, we substitute the given operation into the formula: $$4 ◘ 3 = 4(2) - 3(3) + 1$$ $$= 8 - 9 + 1$$ $$= -1$$ **Conclusion:** Therefore, the value of 4 ◘ 3 is -1. | |||
To find the value of \(4 ◘ 3\), we use the given operation definition \(x ◘ y = 2x - 3y + 1\). Here, \(x = 4\) and \(y = 3\). Substituting these values into the operation's formula gives: \[4 ◘ 3 = 2(4) - 3(3) + 1.\] Doing the arithmetic: \[= 8 - 9 + 1.\] \[= -1.\] Therefore, the value of \(4 ◘ 3\) is \(-1\). | |||
✅ | To find the value of 4 ◘ 3, we substitute x = 4 and y = 3 into the operation ◘: 4 ◘ 3 = 2(4) - 3(3) + 1 4 ◘ 3 = 8 - 9 + 1 4 ◘ 3 = 0 Therefore, the value of 4 ◘ 3 is 0. | ||
✅ | ✅ | Here's how to solve the problem: 1. **Identify the given values:** We are given x = 4 and y = 3. 2. **Substitute the values into the operation definition:** 4 ◘ 3 = (2 * 4) - (3 * 3) + 1 3. **Perform the calculations:** 4 ◘ 3 = 8 - 9 + 1 4. **Simplify:** 4 ◘ 3 = 0 0 | |
✅ | ✅ | Let's break down the operation ◘ and apply it to the given values. The operation is defined as x ◘ y = 2x - 3y + 1. We are asked to find the value of 4 ◘ 3, so we'll substitute x = 4 and y = 3 into the equation. 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 + 1 = 0 So, the value of 4 ◘ 3 is: 0 | |
✅ | We are given the operation ◘ defined as x ◘ y = 2x - 3y + 1. To find the value of 4 ◘ 3, we substitute x = 4 and y = 3 into the definition. This gives us: 4 ◘ 3 = (2 * 4) - (3 * 3) + 1. Simplifying, we get: 4 ◘ 3 = 8 - 9 + 1. Therefore, 4 ◘ 3 = 0. | ||
✅ | To find the value of 4 ◘ 3, we need to substitute x = 4 and y = 3 into the defined operation x ◘ y = 2x - 3y + 1. Performing the calculation: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = 0 Therefore, 4 ◘ 3 = 0. | ||
We have an interesting operation here that shifts numbers around the number line in a unique way. Let's see how it works with 4 and 3. 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1. So, the solution is **−1**. | |||
First, we will substitute the values of x and y into the given formula: x = 4 y = 3 Now, we will plug these values into the formula for the ◘ operation: 4 ◘ 3 = 2(4) - 3(3) + 1 Next, we will solve the expression: = 8 - 9 + 1 Finally, we will simplify the expression: = -1 So, the value of 4 ◘ 3 is -1. | |||
✅ | ✅ | To determine the value of 4 ◘ 3, we will apply the definition of the operation ◘ to the given real numbers. According to the definition, x ◘ y = 2x - 3y + 1. Now, we'll substitute x with 4 and y with 3: 4 ◘ 3 = 2(4) - 3(3) + 1 = 8 - 9 + 1 = -1 + 1 = 0 The value of 4 ◘ 3 is therefore 0. | |
✅ | ✅ | To solve for the value of \( 4 \ ◘ \ 3 \) using the given operation \( \◘ \), we need to follow the definition of the operation. The operation \( x \ ◘ \ y \) is defined as: \[ x \ ◘ \ y = 2x - 3y + 1 \] We are given \( x = 4 \) and \( y = 3 \). Substituting these values into the operation definition, we get: \[ 4 \ ◘ \ 3 = 2(4) - 3(3) + 1 \] Now, perform the calculations step-by-step: 1. Calculate \( 2(4) \): \[ 2 \times 4 = 8 \] 2. Calculate \( 3(3) \): \[ 3 \times 3 = 9 \] 3. Substitute these results back into the expression: \[ 4 \ ◘ \ 3 = 8 - 9 + 1 \] 4. Simplify the expression: \[ 8 - 9 = -1 \] \[ -1 + 1 = 0 \] Thus, the value of \( 4 \ ◘ \ 3 \) is: \[ 0 \] | |
✅ | ✅ | To find the value of 4 ◘ 3, we can plug in x = 4 and y = 3 into the definition of ◘: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, we can simplify: = 8 - 9 + 1 = -1 + 1 = 0 So, 4 ◘ 3 = 0. Answer: 0 | |
✅ | ✅ | Step 1: First, we understand the operation ◘ is defined as x ◘ y = 2x - 3y + 1 for any real numbers x and y. Step 2: To find the value of 4 ◘ 3, we substitute x = 4 and y = 3 into the given operation. Step 3: We calculate 4 ◘ 3 as (2 * 4) - (3 * 3) + 1. Step 4: This simplifies to 8 - 9 + 1. Step 5: Upon further computation, we get 8 - 9 + 1 = 0. Step 6: Therefore, 4 ◘ 3 = 0. Step 7: Thus, the value of 4 ◘ 3 is 0. Answer: 0 | |
✅ | To find the value of 4 ◘ 3, we will substitute x with 4 and y with 3 in the given operation definition: 4 ◘ 3 = 2(4) - 3(3) + 1 Now, we will perform the operations step-by-step: = 8 - 9 + 1 = -1 + 1 = 0 So, the value of 4 ◘ 3 is 0.<|im_end|> | ||
✅ | ✅ | To find the value of 4 ◘ 3, we need to apply the given operation ◘ to the real numbers 4 and 3 according to the definition provided: x ◘ y = 2x - 3y + 1. Step 1: Identify the values of x and y. In this case, x = 4 and y = 3. Step 2: Apply the operation ◘ using the values of x and y. Using the definition x ◘ y = 2x - 3y + 1, we substitute x = 4 and y = 3 into the equation: 4 ◘ 3 = 2(4) - 3(3) + 1 Step 3: Perform the operations within the equation. 2(4) = 8 (multiplication) 3(3) = 9 (multiplication) Now the equation looks like this: 4 ◘ 3 = 8 - 9 + 1 Step 4: Complete the operations to find the final value. 8 - 9 = -1 (subtraction) -1 + 1 = 0 (addition) Therefore, the value of 4 ◘ 3 is 0. The answer is: 0<|im_end|> | |
✅ | To find the value of 4 ◘ 3 using the defined operation x ◘ y = 2x - 3y + 1, we will substitute x with 4 and y with 3 and then calculate the result step by step. Step 1: Substitute x with 4 and y with 3 in the expression 2x - 3y + 1. 2(4) - 3(3) + 1 Step 2: Perform the multiplication. 8 - 9 + 1 Step 3: Perform the addition and subtraction. 8 - 9 + 1 = -1 + 1 = 0 Therefore, the value of 4 ◘ 3 is 0.<|im_end|> | ||
To find the value of 4 ◘ 3, we substitute x = 4 and y = 3 into the expression x ◘ y = 2x - 3y + 1. So, 4 ◘ 3 = 2(4) - 3(3) + 1. Calculating, we get 4 ◘ 3 = 8 - 9 + 1. Simplifying, we find 4 ◘ 3 = -1. Thus, the value of 4 ◘ 3 is -1. |