Problem 23 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1: Substitute Given Values

Start by substituting the values of the variables into the expression. This means replacing each instance of \(x\) with 7, \(y\) with 3, and \(z\) with 9 in the expression \(2x+3z+5y\).

Step 2: Rewrite the Expression

After substitution, the expression \(2x + 3z + 5y\) becomes \(2(7) + 3(9) + 5(3)\). Write this to clearly show your substituted values.

Step 3: Calculate Each Term Separately

Calculate the value of each term: \(2 \times 7 = 14\), \(3 \times 9 = 27\), and \(5 \times 3 = 15\).

Step 4: Add the Results

Add the results from Step 3 together: \(14 + 27 + 15\).

Step 5: Simplify the Sum

Finally, perform the addition: \(14 + 27 = 41\), then \(41 + 15 = 56\). So, the final result of the expression is 56.

Key Concepts

Substitution Method

The substitution method is a handy technique used to evaluate expressions in algebra by replacing variables with their given values. In our problem, we have an expression that involves variables \(x\), \(y\), and \(z\). To simplify or evaluate such an expression, we substitute each variable with the specific numerical values provided. These values are often given, like \(x=7\), \(y=3\), and \(z=9\) in this exercise.

To perform the substitution method effectively:

• Identify each variable present in the algebraic expression.
• Replace every instance of each variable with its corresponding given number.
• Be systematic and careful to avoid mistakes in substitution.

Substitution simplifies the expression into a purely numerical form, allowing the use of arithmetic operations to find its value.

Arithmetic Operations

Arithmetic operations are the basic calculations you perform once the substitution is complete. These operations include addition, subtraction, multiplication, and division. After substituting the variables with their values, the expression \(2x + 3z + 5y\) is rewritten as \(2(7) + 3(9) + 5(3)\).

Let's break down what happens next:

• Multiplication: Each term is computed separately. You multiply \(2\times7\), \(3\times9\), and \(5\times3\) to get the values 14, 27, and 15.
• Addition: After finding the values of each individual term, you sum them up. So, 14 + 27 + 15 results in 56.

Understanding these operations is crucial for solving algebraic expressions as they form the basis of numerical calculations.

Algebraic Expressions

Algebraic expressions are composed of numbers, variables, and arithmetic operations. They offer a way to describe mathematical relationships in a concise form. In our given problem, the expression \(2x + 3z + 5y\) features:

• Constants: Fixed numerals such as 2, 3, and 5.
• Variables: Letters such as \(x\), \(y\), and \(z\) represent unknown values that can change depending on the problem's conditions.
• Operations: Symbols representing arithmetic processes, for instance, addition and multiplication here.

Algebraic expressions are vital in mathematics since they allow us to generalize formulas and calculate outcomes when specific values are substituted in. They serve as a foundation for both simple and complex mathematical problem-solving.