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=-10
The equation has infinite solutions
The equation has one solution x=0
The equation has no solution

Answer 1

Let's solve the equation step by step:

The given equation is:

\[
4x = 3(x + 5) - 15
\]

First, distribute on the right side:

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

Simplifying the right side gives:

\[
4x = 3x
\]

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

\[
4x - 3x = 0
\]

This simplifies to:

\[
x = 0
\]

Now, we will check whether the equation has one, none, or infinite solutions. We found \(x = 0\), so we will check if this is the only solution.

Substituting \(x = 0\) back into the original equation:

\[
4(0) = 3(0 + 5) - 15
\]

Simplifying both sides:

\[
0 = 15 - 15
\]
\[
0 = 0
\]

Since this is a true statement, it confirms that \(x = 0\) is indeed a valid solution.

Thus, the equation has one unique solution:

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