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Chapter 2: Problem 62

Evaluate \(3 x+y\) for the given replacement values. $$ x=7 \text { and } y=-11 $$

Short Answer

Expert verified
The value is 10.

Step by step solution

01

Identify the Expression

The expression we need to evaluate is \(3x + y\).
02

Substitute the Given Values

Substitute \(x = 7\) and \(y = -11\) into the expression. This gives us \(3(7) + (-11)\).
03

Multiply to Simplify

Multiply 3 by 7 to simplify the expression: \(3 \times 7 = 21\).
04

Add to Complete the Calculation

Add \(21\) and \(-11\) to find the final value: \(21 + (-11) = 10\).

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Key Concepts

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

Substitution in Algebra
In algebra, substitution is a crucial process where we replace variables in an expression with their given values. This technique helps to find the exact value of expressions when specific values are known. For instance, if we have an expression like \(3x+y\) and the values of \(x\) and \(y\) are provided, we substitute these values into the expression.

In our exercise, we substitute \(x = 7\) and \(y = -11\) to transform the expression \(3x + y\) into \(3(7) + (-11)\). By doing this, we turn an algebraic expression into a numerical one that can be easily calculated using arithmetic operations.
Basic Arithmetic Operations
Arithmetic operations form the backbone of evaluating expressions after substitution. These operations include addition, subtraction, multiplication, and division, and they help us simplify expressions to find their values.
  • Multiplication: We begin with multiplying the coefficient 3 by the substituted value of \(x\), which is 7, resulting in \(21\).
  • Addition/Subtraction: Next, we perform the addition with and of \(21\) and the negative of \(11\). This is essentially a subtraction (since adding a negative number is like subtracting), giving us a final result of \(10\).
Each step of arithmetic is completed in sequence to ensure the expression is evaluated accurately. Remember to handle each operation one at a time, especially when dealing with signs and operations on negative numbers.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations. These expressions represent mathematical statements that can model real-world situations or be used to find numerical values. In our task, the expression \(3x+y\) consists of two terms:
  • The term \(3x\), which is a product of a number (3) and a variable (\(x\)).
  • The term \(y\), which is also a variable term.
Algebraic expressions are often used to generalize problems and solutions. By substituting specific values into these expressions, as we did, we transition from a general form to a specific numerical evaluation, thereby solving the problem effectively. Understanding how expressions are constructed can help demystify complex equations and promote a deeper understanding of algebraic logic.

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