Problem 18 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$-9 y=27$$

Step-by-Step Solution

Answer

The solution to \(-9y = 27\) is \(y = -3\).

Step 1: Identify the Equation

The given equation is \(-9y = 27\). Our goal is to solve for \(y\).

Step 2: Use the Multiplication Property of Equality

To isolate \(y\), we divide both sides of the equation by \(-9\). The multiplication property of equality allows us to do this operation to both sides while maintaining the equality.

Step 3: Perform the Division

Divide both sides of the equation by \(-9\): \[\frac{-9y}{-9} = \frac{27}{-9}\]This simplifies to \(y = -3\).

Step 4: Verify the Solution

Substitute \(y = -3\) back into the original equation to check: \(-9(-3) = 27\), which simplifies to \(27 = 27\). The solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra that involves finding an unknown value that makes the equation true. In our example, the equation is \(-9y = 27\), where the variable \(y\) is what we need to solve for.

When faced with such an equation, our aim is to isolate the variable on one side of the equation. Here, the multiplication property of equality comes in handy. This property states that you can multiply or divide both sides of an equation by the same non-zero number without changing the equation's validity.

By dividing both sides of the equation by \(-9\), we systematically isolate \(y\). Solving step-by-step, we start with the problem:

• Divide both sides by \(-9\): \(\frac{-9y}{-9} = \frac{27}{-9}\)
• The equation simplifies to \(y = -3\).

This step-by-step approach ensures a clear path to unravel complexities, helping us to find the correct solution for the unknown variable.

Negative Numbers

Dealing with negative numbers can sometimes be tricky, but understanding their behavior is crucial for solving equations like \(-9y = 27\). Here's a simple way to think about them:

Negative numbers are those less than zero, such as \(-1, -3, -9\), etc. When you multiply or divide two negative numbers, the result is positive. This is important because in our example, both the \(-9\) in \(-9y\) and the division by \(-9\) are negative.

Consider these key points when working with negatives:

• Multiplying or dividing two negatives results in a positive number.
• If only one number is negative in a multiplication or division, the result is negative.
• Always take care with sign changes during calculations to avoid mistakes.

Understanding these simple rules helps you manipulate equations more confidently, as in finding that \(y = -3\) is the solution since dividing \(-27\) by \(-9\) gives a positive \(3\).

Mastering negative numbers equips you with a potent tool for approaching algebraic equations fearlessly.

Verification of Solutions

After solving an equation, it is essential to verify the solution is correct. Verifying helps ensure the solution accurately satisfies the original problem, preventing errors from creeping in.

For \(-9y = 27\), once we solve and find \(y = -3\), we need to substitute back into the original equation to check our work. Here's how:

• Replace \(y\) with \(-3\): \(-9(-3) = 27\).
• Calculate the left side: \(27 = 27\).

The equality \(27 = 27\) confirms our solution is correct, as both sides of the original equation match when the solution is substituted.

Verification might seem like an extra step, but it is invaluable. It builds confidence and ensures accuracy, especially when handling more complicated equations where simple errors can easily go unnoticed.