Problem 31 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 6(y+4) $$

Step-by-Step Solution

Answer

Answer: The simplified expression for the given equation is \(6y+24\).

Step 1: Apply the Distributive Property

To apply the distributive property, multiply the number outside the parenthesis with each term inside the parenthesis. In this case, we have \(6(y+4)\). So, we need to multiply 6 by y and 6 by 4. $$ 6(y+4) = 6\cdot y + 6\cdot 4 $$

Step 2: Carry out the Multiplications

Now that we have set up the multiplication, we need to perform the operations to simplify the expression. $$ 6\cdot y + 6\cdot 4 = 6y + 24 $$ Since there are no like terms in the resulting expression, the multiplication is now complete, and the simplified expression is: $$ 6y+24 $$

Key Concepts

Multiplication

Multiplication is an essential math operation that involves finding the total number of objects when you have groups of the same size. In algebra, it plays a crucial role, especially when dealing with expressions and equations.
For expressions inside parentheses, you'll often use multiplication to simplify or expand them. In our example, when you see the expression \(6(y + 4)\), multiplication involves distributing the number 6 to both terms within the parentheses, this process is known as the "distributive property."
This operation is carried out as follows:

• Multiply 6 by \(y\)
• Multiply 6 by 4

This results in the expression \(6y + 24\). Remember, each time you're multiplying a number by a variable or another number, you're essentially scaling up or combining these quantities.

Like Terms

Understanding like terms is key to simplifying algebraic expressions efficiently. Like terms are those that contain the same variable raised to the same power. These terms can be combined to simplify expressions.
In our example \(6y + 24\), let's identify how we might approach like terms. Once you've distributed and multiplied, you'll look to combine terms that share like variables or constants. Here,

• \(6y\) is a term with the variable \(y\)
• 24 is a constant with no variable

Simplification involves adding or subtracting terms that are "alike." However, in the expression \(6y + 24\), because \(6y\) and \(24\) are unlike, they cannot be combined any further.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. They are the foundation for solving equations and understanding how values relate to each other.
In our example expression, \(6(y + 4)\), we're starting with an expression in a factored form. After distributing and simplifying, you end with \(6y + 24\), which is a linear algebraic expression. These expressions often represent a certain relationship or pattern, involving:

• constants (numbers on their own like 24)
• variables (letters that stand for numbers, such as \(y\))
• coefficients (numbers in front of the variables, here 6 in \(6y\))

Working with algebraic expressions involves understanding these components and how to manipulate them, such as expanding, factoring, and simplifying. With practice, this becomes a powerful tool in mathematics and problem-solving.