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Chapter 1: Problem 14

Evaluate each following expression for \(x=2, y=5,\) and \(z=3 .\) \(2 y+5 z\)

Short Answer

Expert verified
The expression evaluates to 25.

Step by step solution

01

Substitute Values

Replace the variables with their given values: substitute 2 for x, 5 for y, and 3 for z in the expression. So, the expression becomes: \(2 \times 5 + 5 \times 3\).
02

Perform Multiplication

First, perform the multiplication for each term. Calculate \(2 \times 5 = 10\) and \(5 \times 3 = 15\). This transforms the expression to: \(10 + 15\).
03

Calculate the Sum

Add the results from the previous multiplications: \(10 + 15 = 25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution in Algebra
Substitution in algebra is a fundamental concept that involves replacing variables in an expression with their numerical values. This process is crucial when you are given a specific value for each variable. It helps in simplifying and evaluating expressions.
In the given problem, you're asked to substitute the values of the variables, where:
  • x = 2,
  • y = 5, and
  • z = 3.
Since the expression provided is \(2y + 5z\), substitution involves replacing \(y\) with \(5\) and \(z\) with \(3\). This will give us the new expression: \(2 \times 5 + 5 \times 3\).
By substituting the values, we prepare the expression for evaluation using arithmetic operations.
Order of Operations
Once substitution is complete, knowing the order of operations is necessary to correctly evaluate an expression. The order of operations is a set of rules that dictate the sequence in which operations should be carried out to ensure consistency.
  • Firstly, calculate operations inside parentheses (though there are none in this problem).
  • Next, perform any exponents, which do not exist here.
  • Then, carry out multiplication and division from left to right.
  • Finally, perform addition and subtraction from left to right.
In our example,\(2 \times 5 + 5 \times 3\), the operations involved are multiplication and addition. According to the order of operations, you must multiply before adding. Calculate each product separately: \(2 \times 5 = 10\) and \(5 \times 3 = 15\), resulting in the simplified expression \(10 + 15\). Finally, you proceed to addition, yielding \(25\).
Basic Arithmetic Operations
Basic arithmetic operations are the building blocks for all math problems. In this expression, you only need to consider multiplication and addition. Let's look at these operations:
  • Multiplication: This operation involves combining groups of equal sizes. For example, multiplying \(2 \times 5\) means adding two sets of five.
  • Addition: This operation totals the sums of numbers. So when adding \(10 + 15\), you're simply calculating the sum of these results.
Both operations are straightforward, but they follow specific rules in the order of operations. Remembering to multiply first and then add ensures you arrive at the correct solution, which in our example equates to \(25\). Understanding these basics make evaluating algebraic expressions much easier.

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