Question

Solve for x to determine if the equation has one, none, or infinite solutions.
4x=3(x+5)-15
The equation has one solution: x=0
The equation has no solution.
The equation has one solution:x=-10
The equation has infinite solutions.

Answer 1

To solve the equation \( 4x = 3(x + 5) - 15 \), we start by simplifying both sides.

First, simplify the right side:

\[
3(x + 5) = 3x + 15
\]
Now substitute back into the equation:

\[
4x = 3x + 15 - 15
\]

This simplifies to:

\[
4x = 3x
\]

Next, subtract \( 3x \) from both sides:

\[
4x - 3x = 0
\]
\[
x = 0
\]

Now let's analyze the result. We found that \( x = 0 \) is a solution.

To also determine the type of solution, consider the steps we took:

If the equation simplifies to a true statement with no variable remaining (like \( 0 = 0 \)), then there would be infinite solutions. If it simplifies to a contradiction (like \( 0 = 5 \)), there would be no solution. In our case, we ended up with one variable solution.

Thus, the final conclusion is:

The equation has one solution: \( x = 0 \).