Problem 35 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 3 x+7 y, \quad x=-1 \text { and } y=-2 $$

Step-by-Step Solution

Answer

The value of the expression is -17.

Step 1: Substitute Values into the Expression

Replace the variable \( x \) with \( -1 \) and the variable \( y \) with \( -2 \) in the expression \( 3x + 7y \). The expression becomes \( 3(-1) + 7(-2) \).

Step 2: Evaluate the Products

Calculate the products individually: \( 3(-1) = -3 \) and \( 7(-2) = -14 \).

Step 3: Sum the Products

Add the results from Step 2: \( -3 + (-14) = -3 - 14 \).

Step 4: Compute the Final Value

Perform the arithmetic operation: \( -3 - 14 = -17 \). Therefore, the value of the expression is \(-17\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. Think of them as codes we use to represent real-life situations using math. An algebraic expression can include:

• Constants: These are specific numbers, like 3 or 7, which don't change.
• Variables: These are symbols or letters, such as \( x \) and \( y \), that represent unknown values and can change.
• Coefficients: These are numbers placed in front of the variables, like 3 in \( 3x \), used to multiply them.
• Operators: These are symbols such as \(+\), \(-\), \(\times\), and \(\div\) that indicate the operations to perform.

Understanding how these components work together is crucial for solving algebraic expressions. By manipulating these parts, you can find the value of the expression when the variables are assigned specific numbers.

Substituting Variables

Substituting variables means replacing the variable with a given number. This step is essential in evaluating algebraic expressions. Here's how you can tackle it:

• Identify the variables: Look at the expression and note which letters are the variables that need values assigned to them.
• Input the values: Replace each variable with the number provided. Be careful to substitute correctly. For instance in \( 3x + 7y \), if \( x = -1 \), replace \( x \) with \(-1\), and if \( y = -2 \), replace \( y \) with \(-2\).
• Transform the expression: Your expression \( 3x + 7y \) becomes \( 3(-1) + 7(-2) \) after substitution.

With substitution done, the expression now contains only numbers, which means it's ready for evaluation through arithmetic operations.

Performing Arithmetic Operations

Arithmetic operations are the basic building blocks of math. They involve simple computations such as addition, subtraction, multiplication, and division. Here's how you use them after substitution:

• Multiplication: Start by multiplying the numbers connected to each variable. In our example, compute \( 3(-1) = -3 \) and \( 7(-2) = -14 \).
• Addition and Subtraction: Once both products are found, perform the addition or subtraction. Add \(-3\) and \(-14\), which means subtracting 14 from -3, to get \(-17\).

Combining these arithmetic steps gives you the expression's complete value. Understanding how to execute these operations accurately ensures correct evaluation of the expression.