Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each algebraic expression for the given value of the variable. $$\frac{3 y-2 y^{2}}{y(y-2)} ; y=5$$

Step-by-Step Solution

Answer

The value of the given algebraic expression when y = 5 is -7/3.

Step 1: Substitute the Given Value

Substitute y = 5 into the algebraic expression, which gives \(\frac{3*5 - 2*5^2}{5*(5-2)}\).

Step 2: Apply BIDMAS Rule

Perform the operations in the proper order: first multiplication, then subtraction in the numerator, and then the operations in the denominator. So we have \(\frac{15 - 2*25}{5*3}\) which simplifies to \(\frac{15 - 50}{15}\). After performing subtraction in the numerator, we get \(\frac{-35}{15}\).

Step 3: Simplify

Simplify the division and get the final result, which is -7/3.

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with their given numerical values. This step is crucial to solving any expression because it allows us to compute actual numerical results. To substitute correctly, identify the variable given in the problem. In our example, this variable is \( y \), and its value is given as 5. Replace every instance of \( y \) in the expression \( \frac{3y - 2y^2}{y(y-2)} \) with 5. This transforms the expression into \( \frac{3\times5 - 2\times5^2}{5(5-2)} \). A common mistake students make is forgetting to substitute all instances of the variable, or missing out the correct order of operations afterward. By methodically replacing each variable, you set a solid foundation for the next steps.

Order of Operations (BIDMAS)

Order of Operations, often remembered by the acronym BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction), dictates the sequence in which operations should be performed to ensure correct results. Let's break it down:

• Brackets: Solve anything inside brackets first. In this case, calculate \( (5-2) \).
• Indices: Next, solve any powers or square operations. Here, compute \( 5^2 \).
• Division and Multiplication: These operations have equal precedence and should be tackled from left to right. Calculate \( 3\times5 \) and \( 2\times25 \), along with any division operations at this stage, which includes \( 5 \times 3 \) in the denominator.
• Addition and Subtraction: Finally, perform addition and subtraction from left to right. Subtraction was involved in simplifying \( 15 - 50 \) in our expression.

In our expression, after applying BIDMAS, we achieved \( \frac{-35}{15} \). Keeping BIDMAS in mind is essential to finding the correct answer.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its lowest terms, which means both the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and use in further calculations or comparisons. Given the expression \( \frac{-35}{15} \), check for common factors between the numerator and denominator. In this case, both numbers are divisible by 5:

• Divide -35 by 5 to get -7.
• Divide 15 by 5 to get 3.

After simplification, the fraction becomes \( \frac{-7}{3} \). It's important to check for divisibility by smaller numbers first, like 2 or 3, if you're struggling to simplify at a glance. Always ensure your final fraction cannot be further simplified to its lowest terms.