If x = 7, y = 4, z = 9, then which of the following is TRUE? I. x + y + z = 20 II. x - y + z = 13 III. x + y - z = 3 IV. -x + y - z = -12 A. I and II B. III and IV C. I and IV D. I and III
C
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.
| Statement | Expression Evaluation | Result | Truth Value |
|---|---|---|---|
| I. \(x + y + z = 20\) | \(7 + 4 + 9 = 20\) | \(20 = 20\) | TRUE |
| II. \(x - y + z = 13\) | \(7 - 4 + 9 = 12\) | \(12 = 13\) | FALSE |
| III. \(x + y - z = 3\) | \(7 + 4 - 9 = 2\) | \(2 = 3\) | FALSE |
| IV. \(-x + y - z = -12\) | \(-7 + 4 - 9 = -12\) | \(-12 = -12\) | TRUE |
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 \(4 - 7\).
Important Questions from Number System
What is the Highest Common Factor of 2 3× 3 5and 3 3× 5 2?
Four prime numbers are arranged in ascending order. The product of the first three numbers is 255 and that of the last three is 1955. The largest prime number is:
Find the number of all prime numbers less than 55.
Value of the square root of \(\frac{36.1}{102.4}\) is:
For any natural number n, 6n - 5n always ends with
