Question
2x+3= 2x+7 5x+5=5x+4(squared) 7x+3=7x+3 3(4x+8)=4(3x+6) (circled) 4x+22=8x+10 3x+18=5x+8(underlined) what do u notice about the circled ones
Answers
Answer
What does it mean to have infinite solutions.
Answered by
GPT-4o mini
Let's analyze the equations you provided:
1. **\(2x + 3 = 2x + 7\)**: This simplifies to \(3 = 7\), which is false. Therefore, it has no solution.
2. **\(5x + 5 = 5x + 4^2\)**: This simplifies to \(5x + 5 = 5x + 16\) or \(5 = 16\), which is also false. So, this has no solution.
3. **\(7x + 3 = 7x + 3\)**: This is true for all values of \(x\). Therefore, it has infinitely many solutions.
4. **\(3(4x + 8) = 4(3x + 6)\)** (circled): This simplifies to \(12x + 24 = 12x + 24\), which is also true for all values of \(x\). So, it has infinitely many solutions.
5. **\(4x + 22 = 8x + 10\)**: This simplifies to \(4x = -12\), or \(x = -3\). So, this has a specific solution.
6. **\(3x + 18 = 5x + 8\)** (underlined): This simplifies to \(2 = 2x\) or \(x = 1\). So, this has a specific solution.
### Observation about the circled equation:
The circled equation \(3(4x + 8) = 4(3x + 6)\) states that both sides are identical expressions, leading to an identity (true for all \(x\)).
In contrast, the other equations tend to either have specific solutions or no solutions. The circled equation is similar to the third equation \(7x + 3 = 7x + 3\) in that they both are identities and exhibit the property of having infinitely many solutions.
Thus, the observation regarding the circled equation is that it is an identity, and inherently, such equations typically have infinitely many solutions, instead of being dependent on the value of \(x\).
1. **\(2x + 3 = 2x + 7\)**: This simplifies to \(3 = 7\), which is false. Therefore, it has no solution.
2. **\(5x + 5 = 5x + 4^2\)**: This simplifies to \(5x + 5 = 5x + 16\) or \(5 = 16\), which is also false. So, this has no solution.
3. **\(7x + 3 = 7x + 3\)**: This is true for all values of \(x\). Therefore, it has infinitely many solutions.
4. **\(3(4x + 8) = 4(3x + 6)\)** (circled): This simplifies to \(12x + 24 = 12x + 24\), which is also true for all values of \(x\). So, it has infinitely many solutions.
5. **\(4x + 22 = 8x + 10\)**: This simplifies to \(4x = -12\), or \(x = -3\). So, this has a specific solution.
6. **\(3x + 18 = 5x + 8\)** (underlined): This simplifies to \(2 = 2x\) or \(x = 1\). So, this has a specific solution.
### Observation about the circled equation:
The circled equation \(3(4x + 8) = 4(3x + 6)\) states that both sides are identical expressions, leading to an identity (true for all \(x\)).
In contrast, the other equations tend to either have specific solutions or no solutions. The circled equation is similar to the third equation \(7x + 3 = 7x + 3\) in that they both are identities and exhibit the property of having infinitely many solutions.
Thus, the observation regarding the circled equation is that it is an identity, and inherently, such equations typically have infinitely many solutions, instead of being dependent on the value of \(x\).
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