For each linear equation, indicate whether the equation has no solution, one solution. or infinitely many solutions.
This problem requires you to show your work.
(3 points)
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6x+2−4x=2(x+2)
8(x+2)=5x+16+3x
3(x+3)=7+3x+6−x
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infinitely many solutions no solution one solution
6x+2−4x=2(x+2):
Simplifying both sides of the equation:
6x + 2 - 4x = 2x + 4
2x + 2 = 2x + 4
Subtracting 2x from both sides:
2 = 4
This is a contradiction, so there is no solution.
8(x+2)=5x+16+3x:
Simplifying both sides of the equation:
8x + 16 = 5x + 16 + 3x
8x + 16 = 8x + 16
Subtracting 8x from both sides:
16 = 16
This equation is always true, so there are infinitely many solutions.
3(x+3)=7+3x+6−x:
Simplifying both sides of the equation:
3x + 9 = 7 + 3x + 6 - x
3x + 9 = 9 + 3x
Subtracting 3x from both sides:
9 = 9
This equation is always true, so there are infinitely many solutions.
Therefore, the answers are:
- For 6x+2−4x=2(x+2), there is no solution.
- For 8(x+2)=5x+16+3x, there are infinitely many solutions.
- For 3(x+3)=7+3x+6−x, there are infinitely many solutions.