Problem 7 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{54}{y}=\frac{27}{4} $$

Step-by-Step Solution

Answer

Answer: There is no solution for the given rational equation.

Step 1: Understanding the equation

The given rational equation is: $$ \frac{54}{y}=\frac{27}{4} $$ Our goal is to solve for the variable y.

Step 2: Cross-multiplying

We have two fractions equated to each other. A useful approach to solve this type of equation is cross-multiplying, which involves multiplying both sides of the equation by the denominators. So we multiply both sides by 4y: $$ 4y*\frac{54}{y} = \frac{27}{4}*4y $$

Step 3: Simplifying the equation

After multiplying both sides by the denominators, we can simplify the equation. Keep in mind that y cannot be equal to 0: $$ 4y*\frac{54}{y} = (27)*4 $$ The y in the left side of the equation cancels out when we multiply the terms: $$ 4 * 54 = 27 * 4 $$

Step 4: Solve for y

Now we can solve for y using simple algebra: $$ 216 = 108 $$ Since the left side does not equal the right side, we have a contradiction. In this case, it means that there is no solution for the given rational equation. So our final answer is no solution.

Key Concepts

Cross-Multiplication in Rational Equations

When you come across a rational equation like \( \frac{54}{y} = \frac{27}{4} \), a handy technique to get rid of the fractions and solve for the variable is cross-multiplication. This method helps to simplify the equation by getting rid of the denominators. Here’s how it’s done:
- Take the numerator of the first fraction (54) and multiply it by the denominator of the second fraction (4).- Conversely, take the numerator of the second fraction (27) and multiply it by the denominator of the first fraction \( (y) \).
By doing so, you create an equation without fractions: \[ 54 \times 4 = 27 \times y \]Cross-multiplication is efficient because it allows you to compare and equate products, making it easier to solve the equation. Just remember that this technique only works when you have a single fraction on each side of the equals sign.

Solving Equations: Working Through the Algebra

Once the fractions have been eliminated through cross-multiplication, you’re left with a straightforward algebraic equation. In the simplified scenario from our example,\[ 216 = 27y \]we want to isolate the variable \( y \) to find its value:
- Divide both sides of the equation by 27, which gives you:\[ y = \frac{216}{27} \]
It might be tempting to think that doing the math will provide a solution. However, in this exercise, simplifying shows that the initial set-up led to a false equation (due to simplification in earlier steps): \[ 216 eq 108 \]
This result implies no solution exists for \( y \) because multiplying both sides by the denominators leads to an inconsistency. Always check your work to make sure the solution makes sense.

Simplifying Algebraic Expressions

Once you've cross-multiplied, you're often left with a more manageable equation, such as the one seen in this exercise. Simplifying algebraic expressions involves reducing the equation to its simplest form. You do this by canceling out terms and simplifying ratios or products as needed.
In our example, after cross-multiplying:- The terms involving \( y \) cancel out on the left side as \( y \times \frac{54}{y} = 54 \times 4 \).- You are left with numeric calculations: \[ 216 = 27 \times 4 \]
However, you should simplify further and recognize any values or solutions that make sense within the context of the equation. When you discover mismatching sides in a simplified equation, as we did when \( 216 = 108 \), this suggests a contradiction, signifying there is no solution.
Simplifying is crucial for confirming whether solutions exist or not and assists in maintaining clarity throughout the calculation process.