Problem 46 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ -4 x+9 y-3 x-y, \quad x=-4 \text { and } y=7 $$

Step-by-Step Solution

Answer

The expression evaluates to 84.

Step 1: Substitute the Values

Replace each occurrence of \(x\) with -4 and each occurrence of \(y\) with 7 in the expression. The expression is \(-4x + 9y - 3x - y\). It becomes \(-4(-4) + 9(7) - 3(-4) - 7\).

Step 2: Simplify Multiplication

Calculate the products for each term: \(-4(-4) = 16\), \(9(7) = 63\), \(-3(-4) = 12\).So the expression becomes \(16 + 63 + 12 - 7\).

Step 3: Combine the Terms

Add and subtract the calculated products: \(16 + 63 + 12 - 7 = 84\).

Key Concepts

Substitution Method

To evaluate an algebraic expression, the substitution method is an essential process. It involves replacing the variables in an expression with given numerical values. In our exercise, we started with the expression \(-4x + 9y - 3x - y\). Here, the variables are \(x\) and \(y\). By substituting the values given in the problem—\(x = -4\) and \(y = 7\)—each instance of \(x\) becomes \(-4\) and each \(y\) becomes \(7\). This step converts the expression directly into numbers as such: \(-4(-4) + 9(7) - 3(-4) - 7\).
Using substitution simplifies our next moves, replacing a potentially confusing arrangement of symbols with manageable arithmetic calculations. It is important to substitute accurately to ensure the integrity of your work in any mathematical context. Practicing this step builds your confidence in handling expressions with multiple variables.

Simplifying Expressions

Once values are substituted into the expression, the next step is to simplify it. Simplifying involves handling operations like multiplication and combination of like terms. For the expression we substituted, \(-4(-4) + 9(7) - 3(-4) - 7\), start with each multiplication:

• Calculate \(-4\times (-4) = 16\)
• Calculate \(9\times 7 = 63\)
• Calculate \(-3\times (-4) = 12\)

The result produces a simpler series of terms: \(16 + 63 + 12 - 7\).
At this stage, the expression is easier to work with because it now consists of addition and subtraction, rather than a mix of operations on variables. Simplifying expressions reduces complexity, aiding clearer and faster calculation.

Evaluating Expressions

The final stage in our process is evaluating the simplified expression. Evaluating means performing the arithmetic operations to arrive at a single numerical value. For our expression, \(16 + 63 + 12 - 7\), perform the operations in sequence:

• Add: \(16 + 63 = 79\)
• Add: \(79 + 12 = 91\)
• Subtract: \(91 - 7 = 84\)

After completing these steps, the evaluated value of the expression is 84.
This process demonstrates how numbers replace variables to yield a concrete result. Evaluating expressions helps us translate mathematical ideas into understandable numerical outcomes. It's crucial because it connects abstract algebra with real-world computations.