Evaluating Algebraic Expressions

This problem requires us to evaluate four different algebraic expressions using given values for the variables \(x\), \(y\), and \(z\). We are given that \(x = 7\), \(y = 4\), and \(z = 9\). We need to substitute these values into each expression and determine which of the resulting equations are true.

Let's evaluate each statement:

Evaluating Statement I: \(x + y + z = 20\)

Substitute the given values into the expression on the left side:

\(x + y + z = 7 + 4 + 9\)

Performing the addition:

\(7 + 4 + 9 = 11 + 9 = 20\)

The equation becomes \(20 = 20\), which is true.

Therefore, Statement I is TRUE.

Evaluating Statement II: \(x - y + z = 13\)

Substitute the given values into the expression on the left side:

\(x - y + z = 7 - 4 + 9\)

Performing the operations:

\(7 - 4 + 9 = 3 + 9 = 12\)

The equation becomes \(12 = 13\), which is false.

Therefore, Statement II is FALSE.

Evaluating Statement III: \(x + y - z = 3\)

Substitute the given values into the expression on the left side:

\(x + y - z = 7 + 4 - 9\)

Performing the operations:

\(7 + 4 - 9 = 11 - 9 = 2\)

The equation becomes \(2 = 3\), which is false.

Therefore, Statement III is FALSE.

Evaluating Statement IV: \(-x + y - z = -12\)

Substitute the given values into the expression on the left side:

\(-x + y - z = -7 + 4 - 9\)

Performing the operations:

\(-7 + 4 - 9 = -3 - 9 = -12\)

The equation becomes \(-12 = -12\), which is true.

Therefore, Statement IV is TRUE.

Summary of Results

Based on our evaluation:

• Statement I: TRUE
• Statement II: FALSE
• Statement III: FALSE
• Statement IV: TRUE

The statements that are true are I and IV.

Matching Results with Options

We need to find the option that states "I and IV". Looking at the provided options, we find that the option corresponding to "I and IV" aligns with our findings.

Conclusion

Statements I and IV are the ones that are true when \(x=7\), \(y=4\), and \(z=9\).

Revision Table: Checking Truth Values

Let's quickly summarize the evaluation for clarity.

Additional Information: Substitution in Algebra

Substitution is a fundamental concept in algebra. It involves replacing variables in an expression or equation with specific numerical values or other algebraic expressions. This process is crucial for:

• Evaluating expressions to find their numerical value for given conditions.
• Checking if a given value or set of values is a solution to an equation or inequality.
• Simplifying complex expressions by substituting known equivalents.
• Solving systems of equations (using the substitution method).

When substituting values, especially negative numbers, it's good practice to use parentheses to avoid errors, particularly with signs and order of operations. For example, if you were evaluating \(y - x\) with \(y=4\) and \(x=-7\), substituting directly without parentheses might lead to \(4 - -7\), which is correct, but writing \(4 - (-7)\) makes the operation clearer and helps prevent mistakes like \(4 - 7\).