Problem 29 - Elementary Algebra

Question

Solve. $$ 27=18 y $$

Step-by-Step Solution

Answer

\( y = \frac{3}{2} \)

Step 1: Understand the Equation

We are given the linear equation \( 27 = 18y \). Our goal is to isolate \( y \).

Step 2: Isolate the Variable

To isolate \( y \), we need to divide both sides of the equation by 18, which is the coefficient of \( y \).

Step 3: Perform Division

Divide 27 by 18 to solve for \( y \): \( y = \frac{27}{18} \).

Step 4: Simplify the Fraction

Simplify \( \frac{27}{18} \) by finding the greatest common divisor (GCD) of 27 and 18, which is 9. Dividing both the numerator and the denominator by 9 gives us \( \frac{3}{2} \).

Step 5: Write the Final Solution

The simplified solution for \( y \) is \( \frac{3}{2} \).

Key Concepts

Solving Equations

Solving linear equations is a fundamental aspect of mathematics that involves finding the value of the unknown variable. In our given exercise, we started with the equation \( 27 = 18y \). This is a straightforward linear equation, where our task is to find the numerical value for \( y \).
Linear equations are typically in the form \( ax + b = c \). However, sometimes additional terms or coefficients may be absent, simplifying the equation structure. In our example, we simply have \( ay = c \), where \( a = 18 \) and \( c = 27 \).
Remember, solving equations hinges on the idea of performing operations equally on both sides to maintain balance, just like a scale.
This can involve:

• Adding or subtracting terms
• Multiplying or dividing by a constant
• Applying inverse operations to simplify the equation

Understanding these principles helps you solve any linear equation more effectively.

Simplifying Fractions

Simplifying fractions is the process of making them as simple as possible. It involves reducing the fraction's numerator and denominator by their greatest common divisor (GCD). In the solution of our equation, we needed to simplify the fraction \( \frac{27}{18} \).
Here are the steps involved in simplifying fractions:

• First, identify the greatest common divisor (GCD) of the numerator and the denominator. In our case, the GCD of 27 and 18 is 9.
• Next, divide both the numerator (27) and the denominator (18) by their GCD (9).
• This gives us the simplified fraction \( \frac{3}{2} \).

Simplifying fractions not only helps in keeping numbers manageable but is also often necessary in presenting the final solution in its most understandable form.
When working with fractions, always look for common factors that can be divided out to ensure your answer is as simplified as possible.

Isolation of Variables

Isolation of variables is a critical step in solving equations, particularly when you aim to solve for a particular variable, like \( y \) in our problem. The essence is to transform the equation in such a way that the target variable is by itself on one side of the equation.
Let's look at our equation \( 27 = 18y \). To isolate \( y \), we needed to address the coefficient (which is 18).
Here is the isolation process broken down:

• Identify the variable you need to solve for. In this case, it's \( y \).
• Determine what operations have been applied to that variable. Here, \( y \) is multiplied by 18.
• Use the opposite operation to isolate the variable. Since \( y \) is multiplied, we divide both sides by 18.
• This leaves us with \( y = \frac{27}{18} \), which we then simplify.

Isolating the variable involves reversing the operations around the variable, which effectively undoes the equation's setup, allowing you to solve for the unknown.