Problem 44 - Algebra for College Students

Question

Solve each equation. $$\frac{y}{3}=\frac{27}{y}$$

Step-by-Step Solution

Answer

y = 9 or y = -9

Step 1 - Multiply both sides by y

To eliminate the fraction on the right side, multiply both sides of the equation by y. This gives: \[\frac{y}{3} \times y = \frac{27}{y} \times y\] Simplifying this, we get: \[\frac{y^2}{3} = 27\]

Step 2 - Multiply both sides by 3

To clear the fraction on the left side, multiply both sides of the equation by 3. This gives: \[\frac{y^2}{3} \times 3 = 27 \times 3\] Simplifying this, we get: \[y^2 = 81\]

Step 3 - Take the square root of both sides

To solve for y, take the square root of both sides of the equation. This gives: \[\frac{y^2}{3} = 27\] Simplifying this, we get: \[y = \frac{\text{y}^2}{27}\]

Key Concepts

algebraic steps

Solving rational equations can seem tricky, but once you get the hang of the steps, it becomes simpler! Let's break down the steps as given in the exercise. First, look carefully at your equation and identify any fractions.
In the given equation, we have \(\frac{y}{3} = \frac{27}{y}\).
Our goal is to isolate the variable (in this case, 'y') so we can solve for its value. To do this, we'll clear out the fractions first. Identify the operations needed, perform them systematically, and simplify the equation at each step till you get to the solution.
Now, let's understand each of these steps in the following sections.

multiplication to eliminate fractions

The first major step in the exercise is to use multiplication to eliminate the fractions. Fractions can make equations look complex, but multiplying can simplify them!
Starting from the original equation \(\frac{y}{3} = \frac{27}{y}\), we want to remove the denominator by multiplying both sides of the equation by 'y'.
This makes the equation: \(\frac{y}{3} \times y = \frac{27}{y} \times y\).
This step results in a new, simplified equation where the fractions are gone: \[ \frac{y^2}{3} = 27 \].
One last multiplication to eliminate the fraction: we multiply both sides by 3 to clear the denominator: \(\frac{y^2}{3} \times 3 = 27 \times 3\).
Now we have removed fractions completely, resulting in: \[ y^2 = 81 \].
We’re left with a quadratic equation! Notice that removing fractions simplifies the path to solving for 'y'.

solving quadratic equations

We are now left with a quadratic equation from our previous steps: \[ y^2 = 81 \]. Quadratic equations often involve a variable raised to the second power. Solving these involves isolating the variable (in this case y) by taking the square root of both sides.
Taking the square root of both sides of \[ y^2 = 81 \], we obtain: \[ y = \pm \sqrt{81} = \pm 9 \].
The '±' sign indicates there are two solutions: y can be both 9 and -9.

• Thus, we consider both positive and negative roots, resulting in: y = 9 and y = -9.

This means the solutions to the original equation \(\frac{y}{3} = \frac{27}{y}\) are y = 9 and y = -9.
This section showcases solving quadratics is a straightforward process once you reduce the equation to a simple form and isolate the term with the variable. Great work following along these steps to solve rational equations!