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When you see an equation such as: 4x+18=4x+5

you can tell how many solutions it will have for x.

Which is a good explanation of how you can tell?

1 answer

To determine how many solutions the equation \(4x + 18 = 4x + 5\) has, you can follow these steps:

  1. Isolate the variable terms: Start by attempting to simplify the equation to see if the variable \(x\) can be isolated. In this case, you can subtract \(4x\) from both sides of the equation:

    \[ 4x + 18 - 4x = 4x + 5 - 4x \]

    This simplifies to:

    \[ 18 = 5 \]

  2. Evaluate the resulting statement: The statement \(18 = 5\) is clearly false. This indicates that there are no values of \(x\) that can satisfy the original equation.

  3. Conclusion: Since the equation leads to a false statement, it means that there are no solutions for \(x\).

In general, if you simplify an equation and end up with a statement that is always true (like \(0=0\)), it indicates there are infinitely many solutions. If you end up with a false statement, as we did here, there are no solutions. If you arrive at a true statement with a variable still present, then there will be exactly one solution.

So, in summary, by simplifying the equation and analyzing the result, you can determine that the equation has no solutions because the resulting statement \(18=5\) is false.