How many solutions does 3x – 1 = 3x + 1 have?
The Correct Answer and Explanation is:
Correct Answer:
The equation 3x – 1 = 3x + 1 has no solution.
Explanation:
To determine how many solutions this equation has, let us solve it step by step:
The given equation is:
3x – 1 = 3x + 1
The first step is to get all the terms involving x on one side and the constants on the other side. We can subtract 3x from both sides:
3x – 1 – 3x = 3x + 1 – 3x
This simplifies to:
–1 = 1
At this stage, we notice that the variable x has been eliminated from the equation, leaving us with:
–1 = 1
This is a false statement because negative one does not equal one.
When solving an equation, if the variables cancel out and the result is a false statement, it means that no values of x will satisfy the equation. Therefore, the equation has no solution.
Why does this happen?
In this particular equation, both sides contain the term 3x, which are identical. Subtracting 3x from both sides removes all variable terms, leaving only constants. If those constants contradict each other, as in this case where –1 ≠ 1, the equation cannot be true for any value of x.
Types of outcomes in linear equations:
There are three possible outcomes when solving a linear equation:
- One unique solution – occurs when you isolate x and find a specific value.
- No solution – occurs when variables cancel and the remaining statement is false.
- Infinite solutions – occurs when variables cancel and the remaining statement is always true, like 0 = 0.
Since we end up with –1 = 1, a false statement, the correct conclusion is:
No solution exists for the equation 3x – 1 = 3x + 1.
