how do you know whether an equation is true, false, or an open sentence?
To determine whether an equation is true, false, or an open sentence, you need to understand the definitions of each:
1. True Equation: If an equation is true, it means that the mathematical statement it represents is correct. When you substitute the values of the variables in the equation, both sides of the equation will be equal.
2. False Equation: If an equation is false, it means that the mathematical statement it represents is incorrect. When you substitute the values of the variables in the equation, both sides of the equation will not be equal.
3. Open Sentence: An open sentence is an equation or inequality that contains one or more variables. It does not become true or false until values are substituted into the variables. Open sentences are not evaluated for truth or falsehood until specific values are inserted.
To determine the nature of an equation, you need to follow these steps:
1. Identify the variables: Determine the variables present in the equation. Variables are typically represented by letters, like x, y, or z.
2. Substitute values: If possible, substitute specific values for each variable in the equation. Make sure to choose values that are meaningful and applicable to the context of the problem.
3. Simplify both sides: Perform the necessary arithmetic calculations on both sides of the equation. Use mathematical operations like addition, subtraction, multiplication, and division to simplify the expression.
4. Compare the results: Compare the simplified expressions on both sides of the equation. If they are equal, then the equation is true. If they are not equal, then the equation is false.
If the equation contains variables but no values have been substituted, it is an open sentence, as it does not have a definitive truth value until you fill in the variables with specific values.
By going through these steps, you can determine whether an equation is true, false, or an open sentence based on the given values or lack thereof.