Problem 72 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$G(a)=\frac{3-5 a}{2 a+7}, G(5)$$

Step-by-Step Solution

Answer

\(G(5) = \frac{-22}{17}\)

Step 1: Understand the Given Rational Expression

The given rational expression is \(G(a) = \frac{3-5a}{2a+7}\). The task is to find the value of this expression at \(a = 5\).

Step 2: Substitute the Value into the Expression

Substitute \(a = 5\) into the expression \(G(a)\). This gives us: \[G(5) = \frac{3 - 5 \cdot 5}{2 \cdot 5 + 7}\]

Step 3: Simplify the Numerator

Calculate the numerator of the expression by substituting \(5\) for \(a\): \[3 - 5 \cdot 5 = 3 - 25 = -22\]

Step 4: Simplify the Denominator

Calculate the denominator of the expression by substituting \(5\) for \(a\): \[2 \cdot 5 + 7 = 10 + 7 = 17\]

Step 5: Compute the Result

Now, substitute the simplified numerator and denominator back into the expression: \[G(5) = \frac{-22}{17}\]

Key Concepts

Substitution

Substitution is a key concept in evaluating rational expressions. It involves replacing variables in an expression with specific values. In our exercise, we began with the rational expression, \(G(a)=\frac{3-5a}{2a+7}\), and we needed to find its value for a specific value of 'a', precisely when \(a=5\).

To apply substitution, we replaced every instance of 'a' in the expression with '5'. This resulted in: \(G(5)=\frac{3-5 \cdot 5}{2 \cdot 5 + 7}\).

This step helps us transform an abstract expression into a concrete numerical form, making it easier to work with. Substitution is crucial when dealing with algebraic expressions and equations.

Numerator and Denominator Simplification

Simplifying the numerator and denominator separately is a fundamental strategy in dealing with rational expressions. This process often involves arithmetic operations like addition, subtraction, multiplication, and division.

For the numerator in our example, we calculated: \(3-5\cdot 5=3-25=-22\). We performed the multiplication first ( \(-5 \cdot 5)\) and then the subtraction ( \(3-25)\). As a result, we got -22.

Similarly, for the denominator, we computed: \(2 \cdot 5+7=10+7=17\). Initially, we multiplied 2 by 5, resulting in 10, followed by adding 7 to yield 17. Verifying each arithmetic step ensures the accuracy of our results.

Simplifying both the numerator and the denominator separately before dealing with the overall expression helps in keeping calculations manageable and minimizes errors.

Rational Expressions

Rational expressions involve fractions where both the numerator and the denominator are polynomials. Understanding how to handle these expressions is critical in algebra.

The given rational expression was: \(G(a)=\frac{3-5a}{2a+7}\). This structure implies we need to pay particular attention to both parts of the fraction.

Rational expressions can be simplified, evaluated, or even factored. In this case, we evaluated the expression by substitution and simplification. Firstly, we substituted 5 for 'a' and simplified each part of the fraction. After simplifying, we placed the resultant values back into the expression to get: \(G(5)=\frac{-22}{17}\).

Understanding rational expressions entails mastering handling polynomials in fraction form and knowing the rules of algebraic operations for simplification.