Problem 34 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ y+12=-12 $$

Step-by-Step Solution

Answer

The value of y, solved by the equation, is -24.

Step 1: Understand the Equation

The provided equation is \( y + 12 = -12 \). This is a simple equation with one variable 'y'.

Step 2: Isolate the variable y

The aim is to find the value of y. We can start by subtracting 12 from both sides of the equation to isolate y on one side. So, the equation becomes \( y = -12 - 12 \)

Step 3: Simplify the Right-Hand Side

After subtraction, the equation simplifies to \( y = -24 \).

Key Concepts

Solving Linear Equations

Linear equations are like puzzles. They usually take the form \( ax + b = c \), with straightforward actions needed to solve them. In the case of the equation \( y + 12 = -12 \), you want to make 'y' the star. This means isolating 'y' on one side.

Here's how it happens:

• You perform the same operation on both sides. Here, you subtract 12 from both sides.
• By doing this, you aim to cancel out the +12 next to 'y.'
• Performing these steps correctly, you unravel the equation into \( y = -24 \).

Through this process, the key is maintaining balance. Always do the same thing on both sides. That way, the equation stays true.

Single-Variable Equations

Single-variable equations, such as \( y + 12 = -12 \), revolve around finding the value of one, lonely variable. In this exercise, 'y' plays that role.

To solve these equations, your primary goal is to isolate the variable you’re solving for. You remove anything that's added or subtracted around it:

• Consider what's tagging along with your variable, in this case, +12.
• Apply an opposite operation. For +12, you subtract 12 from both sides.
• After balancing, you reveal the variable's value: \( y = -24 \).

Single-variable equations often provide an entry point into understanding more complex algebra. They teach you to think logically and methodically.

Basic Algebra Concepts

Algebra lays the foundation for understanding mathematics on a deeper level. At its heart are principles that help in organizing and solving equations systematically. Let's break down a few:

• Equations: These are mathematical statements that assert two expressions are equal.
• Variables: Letters like 'y' in our equation represent numbers we aim to find.
• Operations: Addition, subtraction, multiplication, and division are tools used to manipulate equations. In the example \( y + 12 = -12 \), subtraction helps isolate 'y'.

The beauty of basic algebra is its simplicity and power. Learning to translate a real-world problem into an equation lets you derive meaningful solutions using logical steps. Mastering these concepts equips you with critical thinking and problem-solving skills indispensable in further studies and everyday life.