Question

Solve for x to determine if the equation has one, none, or infinite solutions. 4x=3(x+5)−15(1 point) Responses The equation has infinite solutions. The equation has infinite solutions. The equation has no solution. The equation has no solution. The equation has one solution: x=0. The equation has one solution: x equals 0 . The equation has one solution: x=−10.

Answer 1

To solve the equation \( 4x = 3(x + 5) - 15 \), let's start by simplifying the right side.

1. Distribute \( 3 \) in \( 3(x + 5) \):
\[
3(x + 5) = 3x + 15
\]

2. Now, replace \( 3(x + 5) \) in the equation:
\[
4x = 3x + 15 - 15
\]

3. This simplifies to:
\[
4x = 3x
\]

4. To isolate \( x \), subtract \( 3x \) from both sides:
\[
4x - 3x = 0
\]

5. This simplifies further to:
\[
x = 0
\]

Now, let's analyze the original equation to determine the number of solutions:

Both sides of the equation reduced to an expression that is true for one specific value of \( x \), namely \( x = 0 \). Since there was one variable, and the manipulation did not lead to any contradictions, we conclude:

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

So the response is: **The equation has one solution: \( x = 0 \)**.