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How Many Solutions? Quick Check

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Question
Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−7x=3x+6(1 point)
Responses

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=6.
The equation has one solution: x equals 6 .

The equation has no solution.
The equation has no solution.

The equation has one solution: x=10

1 answer

Let's solve the equation step-by-step to determine how many solutions it has:

Given the equation: \[ 5(2x + 2) - 7x = 3x + 6 \]

  1. Distribute on the left side: \[ 10x + 10 - 7x = 3x + 6 \]

  2. Combine like terms: \[ 3x + 10 = 3x + 6 \]

  3. To isolate the variable, subtract \(3x\) from both sides: \[ 10 = 6 \]

This statement is a contradiction because 10 does not equal 6.

Since we arrived at a contradiction, this indicates that the equation has no solution.

Thus, the final response would be: The equation has no solution.