Problem 11 - Fundamentals of Mathematics

Question

Find the value of each expression. $$5 y^{2}+6 y-11, \text { if } y=-1$$

Step-by-Step Solution

Answer

The value of the expression is \(-12\).

Step 1: Substitute the Variable

Let's replace \( y \) with \( -1 \) in the expression \( 5y^2 + 6y - 11 \). This gives us: \( 5(-1)^2 + 6(-1) - 11 \).

Step 2: Calculate the Squared Term

First, calculate \( (-1)^2 \), which equals 1. Substituting back, the expression becomes \( 5 \times 1 + 6(-1) - 11 \).

Step 3: Multiply Coefficients

Now multiply the coefficients: \( 5 \times 1 = 5 \) and then \( 6 imes (-1) = -6 \). This modifies the expression to \( 5 - 6 - 11 \).

Step 4: Simplify the Expression

Now perform the additions and subtractions: \( 5 - 6 = -1 \), and then \( -1 - 11 = -12 \).

Key Concepts

Substituting Variables

Substituting variables is the first essential step when evaluating algebraic expressions. In algebra, a variable represents an unknown value that can change. To find the value of an expression, replace the variable with the given number.

• The variable in our example is y, and the given value is -1.
• Substitution means everywhere you see y in the expression, replace it with -1.
• This transforms the expression from \(5y^2 + 6y - 11\) into \(5(-1)^2 + 6(-1) - 11\).

By doing this, we turn an algebraic expression into a simpler arithmetic expression, which is a crucial step in solving or simplifying any expression.

Simplifying Expressions

After substituting the variable, simplifying the expression involves performing arithmetic operations step by step. Simplification helps make complex expressions easier to handle and solve.

• Start by solving any powers or exponents in the expression.
• For this problem, calculate \((-1)^2\), which results in 1.
• Then, multiply the coefficients with this result: \(5 \times 1\) and \(6 \times (-1)\).

This operation changes our example expression to \(5 - 6 - 11\).
The main aim of simplification is to reduce the expression step-by-step so that adding or subtracting terms becomes straightforward.

Evaluating Polynomials

Evaluating polynomials is the final step after your expression is completely simplified. This involves performing the arithmetic operations to arrive at a single numeric value.

• Begin with the first two terms in your simplified expression: \(5 - 6\).
• This gives you -1 as both terms are numerically calculated.
• Finally, take this result and solve the last operation, \(-1 - 11\), which equals -12.

Thus, the value of the original polynomial expression, \(5y^2 + 6y - 11\), when \(y = -1\), is -12.
Evaluating polynomials allows us to determine specific values and provides a comprehensive understanding of the polynomial's behavior under different conditions.