Problem 38 - Elementary Algebra

Question

Evaluate. \((y-5)(y+6),\) where \(y=5\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Substitute the value of y

Start by substituting the given value of \( y \) into the expression. We have \( y = 5 \). So the expression \((y-5)(y+6)\) becomes \((5-5)(5+6)\).

Step 2: Simplify the expression inside the brackets

Next, simplify the expressions inside each bracket. This gives us \((5-5) = 0\) and \((5+6) = 11\). Thus, the expression now becomes \( (0) imes (11) \).

Step 3: Perform the multiplication

Finally, multiply the results of the brackets: 0 multiplied by any number is 0. Therefore, \( 0 \times 11 = 0 \).

Key Concepts

Substitution

Substitution is the process of replacing a variable with a given value. It's a simple yet crucial concept in evaluating expressions. When you come across an algebraic expression that involves a variable, you need to substitute the variable with its given value to simplify the expression.
In this exercise, we are given the expression \((y-5)(y+6)\) and told to substitute \(y = 5\). So, wherever we see the letter \(y\) in the expression, we replace it with 5. This transforms the expression \((y-5)(y+6)\) into \((5-5)(5+6)\).
Substitution helps create a numerical expression, allowing us to perform arithmetic operations without the variable. It is the first step towards simplifying and solving an algebraic expression.

Simplification

Simplification involves reducing an expression to its simplest form. This often means combining like terms, removing unnecessary parentheses, or performing basic arithmetic operations.
In our exercise, after substitution, we simplify \((5-5)\) and \((5+6)\).

• For \(5-5\), the operation results in 0 because subtracting a number from itself always gives zero.
• For \(5+6\), the operation results in 11 since adding 6 to 5 yields 11.

After simplifying both expressions inside the parentheses, our overall expression becomes \((0)(11)\).
Simplification prepares the expression for final evaluation through multiplication, making calculations easier and more straightforward.

Multiplication

Multiplication is the arithmetic operation of scaling one number by another. Once substitution and simplification are complete, multiplication is the final step to evaluate the expression.
In algebra, multiplying terms involves combining coefficients and variables, but in this case, we've simplified the expression down to numbers. The expression \((0)(11)\) denotes multiplying 0 by 11.
A crucial point to remember in multiplication is the property that any number multiplied by zero results in zero. Therefore, \(0 \times 11 = 0\). This principle simplifies calculations and is particularly useful in determining the outcome of expressions with zero as a factor.
Always ensure multiplication is done correctly as it forms the basis of finding the solution to the algebraic expressions involved.