Let's start by looking at the two equations you've provided and solving them step by step.
Equation 1: \( 3x + 3 = 7x + 3 \)
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Plug in \( x = 3 \):
- Left side: \( 3(3) + 3 = 9 + 3 = 12 \)
- Right side: \( 7(3) + 3 = 21 + 3 = 24 \)
- They are not equal, so \( x = 3 \) is not a solution.
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Plug in \( x = -2 \):
- Left side: \( 3(-2) + 3 = -6 + 3 = -3 \)
- Right side: \( 7(-2) + 3 = -14 + 3 = -11 \)
- They are not equal, so \( x = -2 \) is not a solution.
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Now let's solve the equation:
- Start with the original equation: \( 3x + 3 = 7x + 3 \)
- Subtract \( 3 \) from both sides: \[ 3x = 7x \]
- Subtract \( 3x \) from both sides: \[ 0 = 4x \]
- Dividing both sides by \( 4 \): \[ x = 0 \]
Equation 2: \( 3(4x + 8) = 4(3x + 6) \)
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Plug in \( x = 3 \):
- Left side: \( 3(4(3) + 8) = 3(12 + 8) = 3(20) = 60 \)
- Right side: \( 4(3(3) + 6) = 4(9 + 6) = 4(15) = 60 \)
- They are equal, so \( x = 3 \) is a solution.
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Plug in \( x = -2 \):
- Left side: \( 3(4(-2) + 8) = 3(-8 + 8) = 3(0) = 0 \)
- Right side: \( 4(3(-2) + 6) = 4(-6 + 6) = 4(0) = 0 \)
- They are equal, so \( x = -2 \) is a solution.
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Now let's solve the equation:
- Start with the original equation: \( 3(4x + 8) = 4(3x + 6) \)
- Distributing on both sides: \[ 12x + 24 = 12x + 24 \]
- This shows that both sides are equal for any value of \( x \). This means every \( x \) is a solution.
Summary:
- For Equation 1 \( 3x + 3 = 7x + 3 \), the solution is \( x = 0 \).
- For Equation 2 \( 3(4x + 8) = 4(3x + 6) \), every \( x \) is a solution, so it has infinitely many solutions.
If we test different values like \( x = 3 \) and \( x = -2 \) in Equation 2, we'll see they can work, but for Equation 1 only \( x = 0 \) is a real solution. Thus, we can have different scenarios: one equation may have one solution while another can have infinitely many solutions.
3 answers