Decide whether each equation has one solution, no solutions, or infinitely many solutions.
2(x – 3) = 2x,
3(y – 3) = 2y – 9 + y,
10x – 2 – 6x = 3x – 2 + x,
4(x + 3) + 2x = x – 8
1. 2(x – 3) = 2x
Expanding and simplifying the equation:
2x – 6 = 2x
Subtracting 2x from both sides:
-6 = 0
The equation is contradictory, so it has no solution.
2. 3(y – 3) = 2y – 9 + y
Expanding and simplifying the equation:
3y – 9 = 2y – 9 + y
Combining like terms:
3y – 9 = 3y – 9
Subtracting 3y from both sides:
-9 = -9
The equation is an identity, so it has infinitely many solutions.
3. 10x – 2 – 6x = 3x – 2 + x
Combining like terms:
4x – 2 = 4x – 2
Subtracting 4x from both sides:
-2 = -2
The equation is an identity, so it has infinitely many solutions.
4. 4(x + 3) + 2x = x – 8
Expanding and simplifying the equation:
4x + 12 + 2x = x – 8
Combining like terms:
6x + 12 = x – 8
Subtracting x from both sides:
5x + 12 = -8
Subtracting 12 from both sides:
5x = -20
Dividing both sides by 5:
x = -4
The equation has one solution.
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To determine whether each equation has one solution, no solutions, or infinitely many solutions, we need to simplify and solve each equation.
1) 2(x – 3) = 2x
First, distribute the 2 to the terms inside the parentheses:
2x - 6 = 2x
Next, move all the x terms to one side of the equation and the constant terms to the other side:
2x - 2x = 6
0 = 6
Since the equation simplifies to 0 = 6, which is not true, there are no solutions to this equation.
2) 3(y – 3) = 2y – 9 + y
First, distribute the 3 to the terms inside the parentheses:
3y - 9 = 2y - 9 + y
Next, combine like terms on both sides of the equation:
3y - 9 = 3y - 9
From here, we can see that the variables and constants on both sides of the equation are equal. This means that no matter what value y is, the equation will always be true. Therefore, this equation has infinitely many solutions.
3) 10x – 2 – 6x = 3x – 2 + x
Combine like terms on both sides of the equation:
4x - 2 = 4x - 2
Both sides of the equation are identical, so no matter what value x is, the equation will always be true. Hence, this equation also has infinitely many solutions.
4) 4(x + 3) + 2x = x – 8
First, distribute the 4 to the terms inside the parentheses:
4x + 12 + 2x = x - 8
Next, combine like terms on both sides of the equation:
6x + 12 = x - 8
Move all the x terms to one side and the constant terms to the other side:
6x - x = -8 - 12
5x = -20
Divide both sides of the equation by 5 to solve for x:
x = -4
This equation simplifies to x = -4, so it has only one solution.
In summary:
- The equation 2(x – 3) = 2x has no solutions.
- The equation 3(y – 3) = 2y – 9 + y has infinitely many solutions.
- The equation 10x – 2 – 6x = 3x – 2 + x has infinitely many solutions.
- The equation 4(x + 3) + 2x = x – 8 has only one solution, x = -4.