Problem 19 - Prealgebra

Question

Solve each equation. $$y-3=-6$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Equation

The equation given is \( y - 3 = -6 \). Our goal is to find the value of \( y \) that makes this equation true.

Step 2: Isolate the Variable

Since we want to solve for \( y \), we need to isolate it on one side of the equation. We can do this by adding 3 to both sides of the equation.

Step 3: Add to Both Sides

Add 3 to both sides:\[ y - 3 + 3 = -6 + 3 \]

Step 4: Simplify the Equation

Simplifying both sides gives:\[ y = -3 \]

Step 5: Verify the Solution

Substitute \( y = -3 \) back into the original equation to ensure the solution is correct:\[ (-3) - 3 = -6 \]The left side simplifies to \(-6\), which equals the right side of the equation, confirming our solution is correct.

Key Concepts

Isolate the Variable

In any equation, isolating the variable is a key step in finding the unknown. When we say "isolate the variable," we mean to move all other terms (numbers or expressions) to the opposite side of the equation. This leaves the variable all by itself on one side, making it easier to solve.
In the equation \( y - 3 = -6 \), our target is to have \( y \) alone on one side. To achieve this, we need to remove the number attached to \( y \). Here, it's \(-3\). By adding 3 to both sides, we effectively cancel out the \(-3\) beside \( y \).
This process of isolating the variable helps streamline the solution and prepares us for applying further algebraic principles. It's important to perform the same operation on both sides to maintain equality.

Addition Property of Equality

The addition property of equality is a fundamental rule in algebra. It states that you can add the same number to both sides of an equation without changing the equation's equality.
For the equation \( y - 3 = -6 \), we utilized this property by adding \( 3 \) to both sides. Here's how it looks:

• Original Equation: \( y - 3 = -6 \)
• Add 3 to both sides: \( y - 3 + 3 = -6 + 3 \)

This operation doesn't change the balance of the equation as both sides undergo the same transformation, ensuring they remain equal.
The addition property helps in effectively eliminating unwanted constants, simplifying equations, and moving terms as needed. It's a critical tool for maintaining equality while manipulating algebraic expressions.

Verify the Solution

Verification is the final critical step in solving any algebraic equation. By verifying, you check that the solution you found truly satisfies the original equation.
For our solution \( y = -3 \), substituting it back into the original equation helps confirm its accuracy. Here's what happens:

• Substitute \( y = -3 \): \(-3 - 3 = -6 \)
• This simplifies to \(-6 = -6 \), which confirms the solution is correct.

Verification is like double-checking your work. It confirms that none of the conditions were overlooked and ensures the steps were correctly executed.
It’s a habit that builds confidence and accuracy, especially in more complex equations, ensuring that the correct value of the variable has been found.