Problem 26 - Prealgebra

Question

Solve each equation. $$y+82=-28$$

Step-by-Step Solution

Answer

The solution is \( y = -110 \).

Step 1: Understand the Equation

We need to find the value of the variable \( y \) in the equation \( y + 82 = -28 \). This is a basic linear equation where \( y \) has been increased by 82 to result in -28.

Step 2: Isolate the Variable

To solve for \( y \), we need to isolate it on one side of the equation. We can do this by performing the inverse operation to remove 82. The inverse operation of addition is subtraction.

Step 3: Subtract 82 from Both Sides

Subtract 82 from both sides of the equation to cancel out the +82. This will give us: \[ y + 82 - 82 = -28 - 82 \].

Step 4: Simplify the Equation

Simplify both sides. On the left-hand side, \( 82 - 82 \) cancels out, leaving us with just \( y \). On the right-hand side, calculate \( -28 - 82 \) which equals \( -110 \). Therefore, \( y = -110 \).

Step 5: Verify the Solution

To ensure the solution is correct, substitute \( y = -110 \) back into the original equation: \( -110 + 82 \). Simplifying this gives \( -28 \), confirming that our solution is correct.

Key Concepts

Inverse Operations

When we look at equations, we often use inverse operations to help us solve them. Understanding inverse operations is crucial because it lets us move terms around in the equation without changing what it equals.

Inverse operations are like opposite actions. If you think about addition and subtraction, they're like best friends doing opposite jobs. Addition is all about putting numbers together, while subtraction takes one number away from another.

• The inverse operation of addition is subtraction.
• The inverse of subtraction is addition.

In our original equation, we had the expression \(y + 82 = -28\). To undo an addition of 82, we perform the inverse operation, which is subtraction. This step is how we keep the equation balanced and isolate the variable. Whenever you're trying to solve an equation, think about what operation is applied to the variable and then perform the inverse operation to remove it.

Isolating Variables

To solve for a variable, like \(y\) in our original equation, we need to isolate it on one side of the equation. Isolating the variable makes it the subject of the equation, which means it's by itself on one side. Having the variable by itself allows us to see its value clearly.

Think of isolating a variable as trying to get \( y \) all alone. If you see "\( y + 82 = -28 \)," focus on getting rid of the numbers surrounding \( y \). The goal is to have something like "\( y = ...\)" because this reveals the number that \( y \) equals.

• Step 1: Look at what's added to or subtracted from the variable.
• Step 2: Use inverse operations to remove them and simplify the equation.

By focusing on these steps, we cleared the 82 from the \( y \) term, which put \( y \) all alone, making the solution much clearer.

Integer Operations

Integer operations involve calculations with whole numbers. When working with integers, we follow basic arithmetic rules that guide us in adding, subtracting, multiplying, and dividing these numbers.

Let’s consider the operation we performed in our solution: \(-28 - 82\). Understanding how integers work helps simplify calculations and solve equations correctly.

• When you subtract a positive integer (like 82), you move left on the number line, making the value more negative.
• For problems involving negative integers, adding them means moving further into the negative, while subtracting a negative shifts right.

In the equation, subtracting 82 from \(-28\) led us to find \(-110\). This concept is key for making sure that every step in solving an equation is accurate.

Addition and Subtraction

Addition and subtraction are foundational arithmetic operations. They're not just for bell peppers and tomatoes at the grocery store! They're critical in solving equations, like our example with \( y + 82 = -28 \).

Equations are like balanced seesaws; whatever you do to one side, you must do to the other to keep them balanced. This is where addition and subtraction come into play, maintaining the balance and ensuring the equality holds.

• Addition combines numbers or increases the value of the variable.
• Subtraction removes numbers or decreases the value of the variable.

In our problem, we needed to "un-add" 82 from both sides by subtracting it. This approach confirms that every step in the equation respects and maintains the equation’s balance throughout the solution process.