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
6x + 2 - 4x = 2(x + 2)
Simplify both sides of the equation:
2x + 2 = 2x + 4
Subtract 2x from both sides of the equation:
2 = 4
Since the equation leads to a contradiction (2 = 4 is false), there is no solution.
8(x + 2) = 5x + 16 + 3x
Distribute on the left side of the equation:
8x + 16 = 5x + 16 + 3x
Combine like terms on the right side of the equation:
8x + 16 = 8x + 16
Subtract 8x from both sides of the equation:
16 = 16
Since the equation is always true (16 = 16 is true), there are infinitely many solutions.
3(x + 3) = 7 + 3x + 6 - x
Distribute on the left side of the equation:
3x + 9 = 7 + 3x + 6 - x
Combine like terms on the right side of the equation:
3x + 9 = 10 + 2x
Subtract 2x from both sides of the equation:
x + 9 = 10
Subtract 9 from both sides of the equation:
x = 1
Since the equation has a consistent solution (x = 1), there is one solution.