Problem 14 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ y+12=5 y-4 $$

Step-by-Step Solution

Answer

The solution is \( y = 4 \).

Step 1: Get all y terms on one side

We start with the equation \( y + 12 = 5y - 4 \). To get all the \( y \) terms on one side of the equation, subtract \( y \) from both sides.

Step 2: Simplify the equation

After subtracting \( y \) from both sides, you have: \( 12 = 4y - 4 \). This simplifies the equation, leaving all \( y \) terms on one side.

Step 3: Isolate the y term

Add 4 to both sides of the equation: \( 12 + 4 = 4y \). This results in \( 16 = 4y \), isolating the term with \( y \) on one side of the equation.

Step 4: Solve for y

Now, divide both sides by 4 to solve for \( y \). You get \( y = \frac{16}{4} \), which simplifies to \( y = 4 \).

Key Concepts

Integer Solutions

When we talk about integer solutions, we mean the solutions to an equation that are whole numbers. They do not have fractions or decimal parts. For the equation given, the integer solution is found when solving the equation results in a whole number for \( y \).
Integer solutions are desirable in many math problems because they are straightforward and easy to interpret. When solving equations, an integer solution often confirms that the problem is defined within the set of integers, ensuring exact and precise answers without potential complications from rounding or approximation.
To identify if an integer solution exists, you'll often solve the equation step by step, ensuring each operation preserves the integrity of whole numbers. If at the end of the simplification and isolation process, the solution to the variable is a whole number, you've successfully found an integer solution.

Isolate Variables

Isolating variables is a crucial step in solving equations. This means getting the variable you're solving for, say \( y \), on one side of the equation all by itself. This process involves rearranging the equation and performing operations that put the variable into a clear position, making it obvious what value it equals.
For example, in the equation \( y + 12 = 5y - 4 \), the first step is to isolate \( y \) by moving all \( y \) terms to one side. This involves subtracting \( y \) from both sides so that you move all other terms to the opposite side.

• Perform operations like addition, subtraction, multiplication, or division as needed.
• The operations must be done on both sides of the equation to maintain balance and equality.
• Ensure that each step moves you closer to isolating the target variable.

By isolating the variable, you're simplifying the process to find out what specific number will satisfy the original equation when substituted back in.

Equation Simplification

Equation simplification is the art of making an equation easier to work with, by reducing it to more manageable terms. Simplifying involves performing operations that combine like terms and reduce fractions or complicated expressions. In simpler equations, like \( y + 12 = 5y - 4 \), simplification helps align terms and focus clearly on isolation.
To simplify equations, follow these guiding principles:

• Combine like terms. These are terms that contain the same variables raised to the same power. Adding or subtracting these simplifies your equation significantly.
• Reduce fractions where applicable, so that calculations are straightforward.
• Perform operations systematically: for example, subtract \( y \) from both sides, leading you to \( 12 = 4y - 4 \), and then add 4 to both sides to simplify further to \( 16 = 4y \).
• The final step often involves reducing to the simplest form, here dividing both sides by 4 to get \( y = 4 \).

Equation simplification paves the way toward solutions that are easier to interpret and verify, ensuring clarity and correctness in problem-solving.