Problem 54 - Elementary Algebra

Question

Solve. $$ 24=6-12 y $$

Step-by-Step Solution

Answer

The solution is \(y = -\frac{3}{2}\).

Step 1: Isolate the variable term

Start by isolating the term that contains the variable on one side of the equation. You currently have \(24 = 6 - 12y\). Subtract 6 from both sides to get \(24 - 6 = -12y\). Simplify the left side: \(18 = -12y\).

Step 2: Solve for y

Now that the variable term \(-12y\) is isolated, solve for \(y\) by dividing both sides of the equation by \(-12\). This gives \(y = \frac{18}{-12}\). Simplify the fraction \(\frac{18}{-12}\) to get \(y = -\frac{3}{2}\).

Step 3: Simplification Sentences

It’s important to ensure your fraction is fully simplified. \(\frac{18}{-12}\) can be reduced by dividing the numerator and the denominator by their greatest common divisor, which is 6, leading to \(\frac{-3}{2}\) as the simplest form.

Key Concepts

Isolating Variables

Isolating variables is the first step when solving linear equations, like the equation given in the exercise. The goal of this step is to rearrange the equation in such a way that the variable of interest, in this case, \( y \), is on one side of the equation.
One way to isolate the variable term is by using addition or subtraction to move constants to the other side. In our example, the equation starts as \( 24 = 6 - 12y \). We want \( y \) to be by itself on one side.

• Subtract 6 from each side to move the constant term to the left side.
• This results in the equation \( 18 = -12y \).

All terms without the variable are now isolated on the other side. Isolating the variable means simplifying future steps because you're dealing directly with solving for that particular variable.

Simplifying Fractions

Once you arrive at a fraction, like \( \frac{18}{-12} \), it's crucial to simplify it. Simplifying fractions helps make equations easier to work with and understand. To simplify a fraction, you divide both the numerator and the denominator by the same number until you can't anymore.

• Check if there is a common factor.
• Here, the numerator is 18 and the denominator is -12.

In this particular example, both numbers are divisible by 6. Dividing both by 6 helps reduce the fraction to \( \frac{-3}{2} \). Always look for the largest number that divides evenly into both the numerator and denominator, known as the greatest common divisor, which leads us to our next point.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when simplifying fractions. It's the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD helps you reduce fractions to their simplest form, which is exactly what makes a solution neat and clear.
For example:

• In \( \frac{18}{-12} \), both 18 and -12 are divisible by 6.
• This makes 6 the GCD, simplifying the fraction to \( \frac{-3}{2} \).

To find the GCD, list the factors of each number and identify the largest one they have in common, or use the Euclidean algorithm for larger numbers. Simplifying with the GCD makes working with fractions and equations far more manageable.