Problem 23 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 24 y-2(6-y)=6(3 y+2) $$

Step-by-Step Solution

Answer

The solution for y in the equation 24y - 2(6 - y) = 6(3y+2) is y = 3.

Step 1: Simplify both sides of the equation

Start by simplifying both sides of the equation by distributing and combining like terms.\[24y - 2(6 - y) = 6(3y + 2) \]Distribute the -2 on the left side and the 6 on the right side of the equation. This results in\[24y - 12 + 2y = 18y + 12 \]Add like terms to simplify further:\[26y - 12 = 18y + 12 \]

Step 2: Re-arrange the equation

Next, aim to get all terms containing y on one side and constant terms on the other side. To do this, subtract 18y from both sides and add 12 to both sides, resulting in:\[26y - 18y = 12 + 12\]After simplifying, the equation becomes:\[8y = 24\]

Step 3: Solve for y

Finally, solve for y by dividing both sides of the equation by 8:\[y = \frac{24}{8}\]Solve to get:\[y = 3\]

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs like addition, subtraction, multiplication, and division. For instance, the equation in our exercise has algebraic expressions on both sides with variables like \( y \) and numbers such as 24 and 2. Understanding the structure of algebraic expressions is crucial.

• Terms: These can be numbers themselves or products of numbers and variables, like \( 24y \) and \( -2(6-y) \).
• Operators: Symbols that tell us what operation to execute on terms, such as +, -, *, and /.

An equation is balanced, meaning what's on one side of the '=' sign is equivalent to what's on the other. Our goal is to isolate the variable \( y \). This involves working with these algebraic expressions to manipulate and simplify them, eventually finding the value of the variable.

Distributive Property

The distributive property is a fundamental concept in algebra. It shows us how to simplify expressions by distributing multiplication over addition or subtraction. In our equation, we need to apply this property to simplify both sides:

• On the left side: Distribute \(-2\) to both \(6\) and \(-y\), which changes the expression to \(-12 + 2y\).
• On the right side: Apply the distributive property by multiplying \(6\) with both \(3y\) and \(2\), leading to \(18y + 12\).

This step reduces the complexity of the equation, allowing us to move onto the next step with a simplified expression. Mastering the distributive property makes it easier to handle more complex equations.

Combining Like Terms

Once we've applied the distributive property, we look for like terms to combine. Like terms are terms that have the same variables raised to the same power. In our equation, you observe that:

• \(24y\) and \(2y\) are like terms. Combining them, we get \(26y\).
• \(-12\) is a constant term that can be moved around as needed.
• On the right side, \(18y\) and \(12\) are already simplified in their respective terms.

By combining like terms, we streamline the equation, moving closer to the solution. The key is gathering all similar terms together to make calculation easier, eventually simplifying our expression to \(8y = 24\), allowing us to solve for \( y \) in the final step.