Problem 37 - Algebra 1

Question

Simplify the variable expression. $$-(-5)(y)(-y)$$

Step-by-Step Solution

Answer

The simplified expression of \(-(-5)(y)(-y)\) is \(-5y^2\).

Step 1: Simplify the expression

In the expression \(-(-5)(y)(-y)\), we need to break down the brackets. In doing so, we follow the rules for multiplication of signed numbers: The product of two negative numbers or two positive numbers is a positive number. The product of one negative number and one positive number is a negative number. After that we can simplify the expression as follows: The result of \(-(-5)\) will be 5, because the minus before the bracket changes the sign.

Step 2: Multiply by variables

The simplified expression is now 5*(y)*(-y). The next step is to multiply the numbers with the variables. The result of \(5 * (y) * (-y)\) will be \(-5y^2\), since multiplying two identical variables results in the variable being squared and any number multiplied by a negative number is negative.

Key Concepts

Multiplying Signed Numbers

Understanding how to multiply signed numbers is essential when simplifying algebraic expressions. Signed numbers are numbers that have a positive or negative sign. The rules for multiplying them are consistent and can be summarized as follows:

• Two positive numbers multiplied together result in a positive number (e.g., \(3 \times 2 = 6\)).
• Two negative numbers multiplied together also result in a positive number (e.g., \((-3) \times (-2) = 6\)). This is because the two negative signs cancel each other out.
• In contrast, if you multiply a positive number by a negative number, the result will be negative (e.g., \(3 \times (-2) = -6\)). The same applies if you multiply a negative number by a positive number (e.g., \((-3) \times 2 = -6\)).

When simplifying expressions, knowing these rules helps you change the sign of a product based on the signs of the factors involved. For example, in the expression \(-(-5)\), the two negatives result in a positive 5.

Variable Expressions

Variable expressions are algebraic phrases that include numbers, variables, and operations. Variables are symbols, often represented by letters like \(x\), \(y\), or \(z\), that stand for unknown values. In expressions, these variables can be manipulated just like numbers.

• When you see \(y \times y\), this is an expression with a variable \(y\) being multiplied by itself.
• Variable expressions can vary in complexity, sometimes involving multiple operations such as addition, subtraction, multiplication, and division.
• The goal is often to simplify these expressions, which might involve combining like terms or applying rules of arithmetic, such as the order of operations.

In our original problem, simplifying the expression involves multiplying variables \(y\) and \(-y\), taking special care to apply the rules of signed numbers as mentioned earlier.

Algebraic Multiplication

When dealing with algebraic multiplication, you multiply both numbers and variables. This involves applying the distributive property and rules of exponents when necessary. Multiplication of variables follows these principles:

• If two variables are the same and are multiplied, like \(y \times y\), they become squared (\(y^2\)).
• Any number, whether positive or negative, when multiplied by a variable, requires application of the rules of signed number multiplication, determining the final sign of the product.
• When constants (numbers) and variables are multiplied together, treat the numbers and the letters separately and then combine them in the result.

In our exercise example, after recognizing that \(-(-5)\) simplifies to 5, we multiply this positive number with \(y\) and \(-y\), leading to \(-5y^2\) due to \(y \times -y\) becoming \(-y^2\). The constant \(5\) remains and contributes to the final result.