Problem 24 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$5 z-3 x-2 y$$

Step-by-Step Solution

Answer

The expression evaluates to 18.

Step 1: Substitute Values

Replace the variables in the expression with the given values: \(x = 7\), \(y = 3\), and \(z = 9\). The expression becomes \(5(9) - 3(7) - 2(3)\).

Step 2: Multiply Terms

Calculate the products of the numbers and their coefficients:- \(5 \times 9 = 45\)- \(3 \times 7 = 21\)- \(2 \times 3 = 6\)The expression is now \(45 - 21 - 6\).

Step 3: Perform Subtractions

Subtract the numbers from left to right:- Start with \(45 - 21 = 24\)- Then, \(24 - 6 = 18\)

Step 4: Conclusion

The value of the expression when substituting \(x = 7\), \(y = 3\), and \(z = 9\) is \(18\).

Key Concepts

Substitution Method

The substitution method is a handy tool when evaluating expressions with variables. First, identify the specific values given for each variable in the problem. By substituting these values into the expression, you replace each variable with its corresponding numerical value. For instance, in our original exercise:- We have the variables: \(x = 7\), \(y = 3\), and \(z = 9\).- Substitute these into the expression \(5z - 3x - 2y\), which transforms it into \(5(9) - 3(7) - 2(3)\).This replacement step sets the stage for solving the expression numerically. When variables are replaced by numbers, it simplifies the problem significantly, turning it into straightforward arithmetic operations. This step is crucial as it helps in systematically transitioning from an algebraic expression to a purely numerical calculation.

Order of Operations

Once you substitute the variables, the next logical step is applying the order of operations, often remembered by the acronym PEMDAS:- **P**arentheses- **E**xponents- **M**ultiplication- **D**ivision- **A**ddition- **S**ubtractionWhen working with the expression \(5(9) - 3(7) - 2(3)\), you'll start by performing any multiplication or division in sequence from left to right: - Calculate \(5 \times 9 = 45\),- \(3 \times 7 = 21\),- \(2 \times 3 = 6\).After dealing with the multiplication, handle any addition or subtraction again from left to right:- Begin with \(45 - 21 = 24\),- Then, \(24 - 6 = 18\).Following this order ensures everyone arrives at the same solution, promoting consistency across multiple calculations.

Arithmetic Simplification

Arithmetic simplification involves reducing expressions step by step to their simplest form. After substituting variables and applying the correct order of operations, you address any remaining calculations. Simplification means processing basic arithmetic operations such as addition, subtraction, multiplication, and division to reach a definitive answer.In the evaluated expression \(5(9) - 3(7) - 2(3)\):- You've already completed the multiplication: - \(45\), \(21\), and \(6\) results.- Proceed with subtraction: - \( 45 - 21 = 24\) - Then, simplify further with \( 24 - 6 = 18\).Each simplification stage removes complexity, ensuring that you're left with the simplest version of the expression, which is easy for anyone to understand and verify. Simplification is paramount because it prepares the expression for direct comparison with other values or expressions.