Problem 20 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$12(y+2)+5=2 y-1$$

Step-by-Step Solution

Answer

The solution is \( y = -3 \).

Step 1: Distribute and Simplify

First, distribute the 12 across the parentheses in the left part of the equation.\[ 12(y + 2) = 12y + 24 \]So the equation becomes:\[ 12y + 24 + 5 = 2y - 1 \]Now combine the like terms on the left side of the equation:\[ 12y + 29 = 2y - 1 \]

Step 2: Move variable terms to one side

Next, get all the terms that contain \(y\) on one side. To do this, subtract \(2y\) from both sides of the equation:\[ 12y - 2y + 29 = -1 \]This simplifies to:\[ 10y + 29 = -1 \]

Step 3: Isolate the variable term

Subtract 29 from both sides to start isolating the variable term:\[ 10y + 29 - 29 = -1 - 29 \]This simplifies to:\[ 10y = -30 \]

Step 4: Solve for the variable

Finally, divide both sides by 10 to solve for \(y\):\[ \frac{10y}{10} = \frac{-30}{10} \]Simplifying gives:\[ y = -3 \]

Key Concepts

Understanding the Distributive Property

In solving linear equations, the distributive property is a key tool. It's like spreading out a multiplication over a sum or difference within parentheses.
This makes the equation more straightforward. Consider the equation given:

• 12(y+2) + 5 = 2y - 1

By applying the distributive property, you multiply 12 by both terms inside the parentheses, transforming the expression into:

• 12y + 24

Now, our equation becomes:

• 12y + 24 + 5 = 2y - 1

The complexity of the equation reduces, setting the stage for the next steps of solving it.

Isolating the Variable: The Art of Simplicity

Isolating the variable involves getting all the terms with the variable on one side of the equation. It's like clearing the path to find out what the variable stands for.
Using our current equation, 12y + 29 = 2y - 1, we need to move the "2y" from the right to the left side. This is achieved by subtracting 2y from both sides.

• 12y - 2y + 29 = -1

This simplifies to:

• 10y + 29 = -1

Now, the terms with "y" are isolated to one side, making it a simpler equation and easing the road to finding the value of y.

Combining Like Terms: Simplifying Made Easy

Combining like terms is a way to simplify an equation, merging similar elements to make the math more manageable.
Let's revisit our distributed equation:

• 12y + 24 + 5 = 2y - 1

On the left side, we have two numbers without variables: 24 and 5. Add them together to simplify:

• 12y + 29 = 2y - 1

In math, simplifying where possible is crucial. It makes solving equations faster and more efficient by reducing unnecessary complexity.

Intro to Prealgebra

Prealgebra lays the foundation for more advanced math. It introduces the basic tools and techniques used in algebra, such as equations, variables, and basic arithmetic operations.
In this exercise, we used several key prealgebra concepts:

• The distributive property allowed us to remove parentheses effectively.
• Combining like terms helped simplify our equation significantly.
• Isolating the variable was crucial for finding its value.

These skills are essential building blocks for anyone moving on to more complex algebraic problems. Mastering them in prealgebra equips students for success in later math courses.