Does the equation 5(x−1)=5x−3 have one solution, no solutions, or infinitely many solutions?

Question

First, let’s try to isolate the variable, x. One of the x terms is inside the parentheses on the left. So, let’s start by getting rid of the parentheses to make this equation easier to work with. One way to do this is to distribute the 5 to the x and to the 1. Start by distributing the 5 to the x. How can you show 5·x using an expression?
5(x−1)
=
5x−3

− ?
=
5x−3

Answer 1

To show the distribution of \(5\) to the expression \((x - 1)\), you'll multiply each term within the parentheses by \(5\). Here's how the equation breaks down step by step:

1. Distribute the \(5\) to both \(x\) and \(-1\): \[ 5(x - 1) = 5 \cdot x - 5 \cdot 1 \] This simplifies to: \[ 5x - 5 \]

2. Now, you can rewrite the original equation: \[ 5(x - 1) = 5x - 3 \] becomes: \[ 5x - 5 = 5x - 3 \]

3. Next, let's isolate the variable \(x\). Subtract \(5x\) from both sides: \[ 5x - 5 - 5x = 5x - 3 - 5x \] This simplifies to: \[ -5 = -3 \]

4. The equation \(-5 = -3\) is a contradiction; it is false and indicates that there are no solutions to the original equation.

In conclusion, the equation \(5(x - 1) = 5x - 3\) has no solutions.