Decide whether the equation has one solution, no solution or infinitely amount of solutions.
1: 2(x - 3) = 2x
one
none
infinitely
2: 3(y - 3) = 2y - 9 +y
one
none
infinitely
3: 10x - 2 - 6x = 3x - 2 + x
one
none
infinitely
4(x + 3) + 2x= x - 8
one
none
infinitely
We'll be glad to check your answers.
one
To determine whether each equation has one solution, no solution, or infinitely many solutions, we can solve each equation step by step. Let's go through them one by one:
1: 2(x - 3) = 2x
To solve this equation, we'll need to distribute the 2 on the left side:
2x - 6 = 2x
Notice that we have 2x on both sides of the equation. In this case, we subtract 2x from both sides to isolate the constant term:
-6 = 0
Since this results in a false statement (-6 is not equal to 0), we conclude that there are no solutions to this equation: none.
2: 3(y - 3) = 2y - 9 + y
Let's start by distributing the 3 on the left side:
3y - 9 = 2y - 9 + y
Now, we can combine like terms on both sides of the equation:
3y - 9 = 3y - 9
Observe that we have 3y on both sides. If we subtract 3y from both sides, we are left with:
-9 = -9
This equation is true, which means that it is an identity. In other words, any value of y will satisfy this equation. Therefore, there are infinitely many solutions.
3: 10x - 2 - 6x = 3x - 2 + x
To solve this equation, let's simplify both sides and combine like terms:
10x - 2 - 6x = 3x - 2 + x
4x - 2 = 4x - 2
Notice that we have 4x on both sides of the equation. Subtracting 4x from both sides results in:
-2 = -2
This equation is true because -2 is equal to -2. Therefore, it is an identity, and there are infinitely many solutions.
4(x + 3) + 2x = x - 8
Let's simplify and solve this equation step by step:
4x + 12 + 2x = x - 8
Combine like terms on both sides:
6x + 12 = x - 8
Subtract x from both sides to isolate the constant term:
5x + 12 = -8
Next, subtract 12 from both sides:
5x = -20
Finally, divide both sides by 5:
x = -4
We have obtained a specific value for x, which means there is one unique solution to this equation: one.
In summary:
1: No solution (none)
2: Infinitely many solutions (infinitely)
3: Infinitely many solutions (infinitely)
4: One solution (one)