Problem 43 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(x-3(x-y)\) (a) \(x=3, y=3\) (b) \(x=4, y=-4\)

Step-by-Step Solution

Answer

(a) The value of the expression for x=3, y=3 is 3. (b) The value of the expression for x=4, y=-4 is -20.

Step 1: Substitution for (a)

Substitute the given values into the expression: \(x-3(x-y) = 3 - 3(3-3)\)

Step 2: Simplify for (a)

Simplify the equation by performing the operations: \(3 - 3(0) = 3 - 0 = 3\)

Step 3: Substitution for (b)

Now substitute the second pair of values into the expression: \(x-3(x-y) = 4 - 3(4 - (-4))\)

Step 4: Simplify for (b)

Simplify the equation by performing subtraction and addition operations: \(4 - 3(8) = 4 - 24 = -20\)

Key Concepts

Substitution Method

When evaluating algebraic expressions, the substitution method is an essential technique that helps in replacing variables with their numerical values. In practice, you first identify the variables within the expression and then replace them with the given numbers. For example, in the expression \(x-3(x-y)\), the variables \(x\) and \(y\) are each placeholders for specific values.

Once you've substituted all variables with their respective values, as in \(3 - 3(3-3)\) for \(x = 3\) and \(y = 3\), the expression is ready to be simplified. This method allows you to transition from an abstract algebraic expression to a concrete numerical problem that can be solved using basic arithmetic operations.

In short, the substitution method is a way to make the abstract concrete. It takes the general form and provides a specific instance that is easy to compute, making it one of the foundational skills in algebra.

Simplifying Expressions

Simplifying expressions is a critical step in evaluating algebraic expressions. It involves carrying out addition, subtraction, multiplication, division, and other operations to reduce the expression to its simplest form. The goal is to make the expression as straightforward as possible without changing its value.

Consider the example of simplifying \(3 - 3(0)\). Here, multiplication within the parentheses is performed first, according to the order of operations, yielding zero. Subsequently, three minus zero leaves us with three — the simplified form of the expression. Simplification is not just about making calculations easier; it also helps in better understanding the structure and relationships in algebraic expressions.

To simplify effectively, remember the order of operations (PEMDAS/BODMAS), combine like terms, and factor when possible. By mastering simplifying techniques, you're better equipped to handle complex algebraic challenges.

Expression Evaluation with Variables

Expression evaluation with variables requires a combination of the substitution method and simplification. Variables in an expression stand in for unknowns or quantities that can change, and replacing these with actual values is the first step toward finding an expression's overall value.

Consider our text example for \(x=4, y=-4\). Substitution would yield \(4 - 3(4 - (-4))\), embedding negative signs and subtraction within a multiplication context. Carefully applying the operations—and keeping track of sign changes, which is critical in handling subtraction and addition of negative numbers—is necessary for accurate simplification, bringing us to \(4 - 24 = -20\).

Successfully evaluating expressions with variables hinges on attentiveness to detail and a systematic approach to applying arithmetic operations. With practice, the process becomes intuitive, allowing for seamless navigation through more demanding algebraic problems.