Problem 18 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ -9 y=54 $$

Step-by-Step Solution

Answer

The solution is y = -6, verified by substitution.

Step 1: Isolate the Variable

To solve the equation \(-9y = 54\), divide both sides by \(-9\) to isolate \(y\). This gives:\[y = \frac{54}{-9}\]

Step 2: Simplify the Fraction

Simplify the fraction obtained by dividing 54 by \(-9\):\[y = -6\]

Step 3: Verify the Solution

Substitute \(y = -6\) back into the original equation to ensure it holds true.Check: \(-9(-6) = 54\), which simplifies to 54 = 54. This confirms that the solution is correct.

Key Concepts

Isolating Variables

When solving equations, one crucial step is to isolate the variable you are trying to find. In the exercise, we're working with the equation \(-9y = 54\). The goal is to have \(y\) by itself on one side of the equation. To achieve this, you have to perform the opposite operation of what's being done to the variable.

• Here, \(y\) is being multiplied by \(-9\). To isolate \(y\), divide both sides of the equation by \(-9\).
• This operation gives us the new equation: \(y = \frac{54}{-9}\).

After dividing both sides, the variable \(y\) is isolated on one side of the equation. This makes it easier to solve for its value. Always remember: whatever you do to one side of the equation, do it to the other to maintain balance.

Simplifying Fractions

Once the variable is isolated, the next step is to simplify any fractions that may have been created in the process. In our equation, we ended up with the fraction \(\frac{54}{-9}\). Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor.

• First, identify the greatest common divisor of 54 and 9, which is 9.
• Next, divide both the numerator (54) and the denominator (-9) by 9.
• This gives you the simplified fraction \(\frac{54 \div 9}{-9 \div 9} = \frac{6}{-1} = -6\).

Simplifying helps in finding the clean and straightforward value for \(y\), which in this case is \(y = -6\). It makes working with numbers clearer and reduces potential errors in further calculations.

Checking Solutions

Verifying your solution is a critical step in solving equations. This ensures that the value you found for the variable actually satisfies the original equation. To check if \(y = -6\) is correct, substitute it back into the original equation \(-9y = 54\) and see if both sides of the equation are equal.

• Substitute \(y = -6\) back into the equation: \(-9(-6)\).
• Perform the multiplication: \(-9 \times -6 = 54\).
• Since \(54 = 54\), the equation holds true.

Confirming the solution like this is very important. It ensures accuracy and helps verify that no mistakes were made during the calculation process. Always remember to check your solutions to avoid errors in solving equations.