Problem 37 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{y}{4 \cdot 11}=2.3 $$

Step-by-Step Solution

Answer

Answer: The value of y is $101.2$.

Step 1: Identify the given equation

We are given the following conditional equation: $$ \frac{y}{4 \cdot 11}=2.3 $$

Step 2: Simplify the denominator

Before solving for y, let's simplify the denominator by multiplying 4 and 11: $$ \frac{y}{44}=2.3 $$

Step 3: Isolate y by multiplying both sides by the denominator

To isolate y, we need to eliminate the fraction. We can do this by multiplying both sides of the equation by the denominator, which is 44: $$ 44 \cdot \frac{y}{44} = 2.3 \cdot 44 $$

Step 4: Simplify the equation

After multiplying both sides, the equation simplifies to: $$ y = 101.2 $$

Step 5: Write the answer

We have successfully solved the conditional equation, and the resulting value of y is: $$ y = 101.2 $$

Key Concepts

Conditional Equations

A conditional equation is a type of equation that holds true only for specific values of the variable. Unlike identities, which are true for all values of the variable, conditional equations are valid under certain conditions or assumptions.
When solving a conditional equation, you are looking for the particular solution or solutions that satisfy the equation.
For example, in the equation \( \frac{y}{44} = 2.3 \), the solution is not universally applicable for every \( y \), but rather just when \( y = 101.2 \).
In many math problems, identifying that you are dealing with a conditional equation is the first step to finding the correct solution. Take note of the structure of the equation and the operation involved, as this will guide your approach to finding the solution.

Simplifying Fractions

Simplifying fractions is an essential part of solving equations involving fractions. It involves reducing the fraction to its simplest form by either combining terms in the numerator and the denominator or canceling out common factors from both.
This can make the equation easier to work with.
In our example, we initially have \( \frac{y}{4 \cdot 11} = 2.3 \). Here, simplifying means performing the multiplication in the denominator.

• Firstly, multiply 4 and 11 to get 44.
• The equation then becomes \( \frac{y}{44} = 2.3 \).

This step of simplifying helps in making it easier to perform further algebraic operations, like isolating the variable.

Isolating Variables

Isolating variables is a key step in solving equations, whether they are linear or otherwise.
The goal is to get the variable on one side of the equation, usually the left, and a constant or an expression on the other side. This allows you to find the value of the variable that makes the equation true.
To isolate \( y \) in the equation \( \frac{y}{44} = 2.3 \), follow these steps:

• Recognize that \( y \) is currently divided by 44. To remove this, multiply both sides by 44.
• This gives us \( 44 \cdot \frac{y}{44} = 2.3 \cdot 44 \).
• On the left side, the 44s cancel out because \( 44 \div 44 = 1 \), simplifying it to \( y \).
• On the right side, calculate \( 2.3 \cdot 44 \) to get 101.2.

Therefore, the value of \( y \) that solves the equation is \( y = 101.2 \). Isolating variables can involve various operations such as addition, subtraction, multiplication, or division, depending on what needs to be undone to solve for the variable.