Solve for x to determine if the equation has one, none, or infinite solutions

Question

11x=3(7x-1)-10x

1. the equation has one solution: x=-1
2. The equation has infinite solutions
3. the equation has no solutions
4. the equation has one solution: x=-3

Answer 1

To solve the equation \( 11x = 3(7x - 1) - 10x \), we start by simplifying the right side.

First, distribute the \( 3 \):

\[ 3(7x - 1) = 21x - 3 \]

Now we rewrite the equation:

\[ 11x = 21x - 3 - 10x \]

Combine the \( 21x \) and \( -10x \) on the right side:

\[ 11x = (21x - 10x) - 3 \]

This simplifies to:

\[ 11x = 11x - 3 \]

Next, subtract \( 11x \) from both sides:

\[ 11x - 11x = 11x - 11x - 3 \]

This leads to:

\[ 0 = -3 \]

Since \( 0 = -3 \) is a contradiction, the equation has no solutions.

Thus, the correct option is:

3. the equation has no solutions.

Answer 2

Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)-4=2x+6

1. the equation has infinite solutions
2. the equation has one solution: x=1
3. the equation has no solutions
4. the equation has one solution: x=0

Answer 3

Let's solve the equation \( 5(2x + 2) - 4 = 2x + 6 \).

First, distribute the \( 5 \) on the left side:

\[ 5 \cdot 2x + 5 \cdot 2 - 4 = 2x + 6 \] \[ 10x + 10 - 4 = 2x + 6 \]

Now simplify the left side:

\[ 10x + 6 = 2x + 6 \]

Next, let's get all terms involving \( x \) on one side and constant terms on the other side by subtracting \( 2x \) from both sides:

\[ 10x - 2x + 6 = 6 \] \[ 8x + 6 = 6 \]

Now, subtract \( 6 \) from both sides:

\[ 8x = 0 \]

Finally, divide both sides by \( 8 \):

\[ x = 0 \]

Since we found a specific value for \( x \) (which is \( 0 \)), the equation has one solution.

Thus, the correct option is:

4. the equation has one solution: x=0.