You can identify an equation with infinitely many solutions by attempting to solve it and observing a unique outcome: a variable or number equaling itself. According to the provided reference, this occurs when "any value for the variable makes the equation true."
Identifying Infinite Solutions
When solving an equation, you typically aim to find a specific value for the variable that makes the equation true. However, in cases with infinite solutions, the process leads to a different result.
- Variable Equals Itself: For example, if after simplifying the equation, you end up with x = x, this indicates an infinite number of solutions. It means that no matter what number you plug in for x, the equation will always be true.
- Number Equals Itself: Similarly, if the equation simplifies to a true numerical statement, such as 5 = 5, this also signals an infinite number of solutions. This again indicates the equality holds true regardless of the variable's value.
Practical Example
Let's consider the equation:
2(x + 3) = 2x + 6
Let's solve it step by step:
- Distribute the 2 on the left: 2x + 6 = 2x + 6
- Subtract 2x from both sides: 6 = 6
Notice that we are left with a true numerical statement (6 = 6), meaning there are infinite solutions. Any value of x will make the equation true.
Table Summary
| Condition | Result | Explanation | Solutions |
|---|---|---|---|
| Variable = Variable | x = x | Any value of x makes the equation true. | Infinite |
| Number = Number | 5 = 5 | The equality holds true, regardless of the variable's value. | Infinite |
| Variable = Number | x = 5 | Only one specific value of x makes the equation true. | Unique |
Key Takeaway
The presence of a variable equaling itself (like x = x) or a numerical equality (like 5 = 5) when solving an equation definitively indicates that the equation has an infinite number of solutions. These results reveal that any value of the variable satisfies the original equation.
