Problem 13 - Precalculus

Question

Solve each equation. $$1-\frac{y}{3}=6$$

Step-by-Step Solution

Answer

The solution is \( y = -15 \).

Step 1: Transposing the Fraction

First, we need to move the fractional term to the other side of the equation. Start by subtracting 1 from both sides: \\( 1 - \frac{y}{3} - 1 = 6 - 1 \) \This simplifies to: \\( -\frac{y}{3} = 5 \)

Step 2: Clearing the Negative Sign

Multiply both sides of the equation by -1 to clear the negative sign: \\( \frac{y}{3} = -5 \)

Step 3: Isolating y

To isolate \(y\), multiply both sides by 3 to eliminate the fraction: \\( y = -5 \times 3 \) \Simplifying gives \( y = -15 \).

Key Concepts

Fractional Equations

Fractional equations are equations that include fractions containing variables in the numerator, the denominator, or both. When faced with a fractional equation, one of the primary goals is to eliminate the fractions to simplify the solving process.
A practical way to approach these equations often involves finding a common denominator or directly eliminating the fractions by multiplying each part of the equation by the denominator itself.

• This removal helps simplify the equation into a more manageable form—usually a linear equation.
• In this exercise, the given term \(-\frac{y}{3}\) is a fractional term that required elimination.

This leads us into our next concept of transposing terms to effectively manage the equation.

Transposing Terms

Transposing terms is a fundamental skill in algebra, essential for solving equations. It involves rearranging terms of an equation to isolate a specific variable. When transposing, what you do on one side of the equation, you must do on the other to maintain equality.
In the example equation, \(1 - \frac{y}{3} = 6\), the goal was to gradually isolate the variable.

• The initial step involved subtracting 1 from both sides, allowing us to clear any terms not associated with the variable \(y\).
• This resulted in a simplified equation \(-\frac{y}{3} = 5\), setting us up for the next necessary adjustments.

Transposing terms sets the basis for further operations, such as multiplying equations to remove any negative signs or fractional factors.

Multiplying Equations

Multiplying equations is an essential technique used particularly when dealing with equations that have negative or fractional components. This method allows for the elimination of undesired signs or factors to further simplify the equation.
In this specific problem, once the equation was simplified to \(-\frac{y}{3} = 5\), multiplying both sides by \(-1\) eradicated the negative sign in the fractional component:

• It transformed the equation into \(\frac{y}{3} = -5\), now free of any negative implications and readier for further simplification.
• Multiplying each factor of the equation helps maintain the balance, ensuring the equation remains true throughout the simplification process.

This necessary manipulation ensures an easier path towards isolating the variable in subsequent steps.

Variable Isolation

Variable isolation is the final step in solving the linear equation, focusing on leaving the variable alone on one side of the equation. This step involves removing all coefficients or factors attached to the variable.
For our solved equation, \(\frac{y}{3} = -5\), the goal was to eliminate the fraction completely:

• This was achieved by multiplying both sides of the equation by the denominator, which was 3.
• The resulting equation \(y = -5 \times 3\) simplifies directly to \(y = -15\), revealing the solution where the variable \(y\) stands alone.

Mastering variable isolation in linear equations equips you with the ability to decipher even more complex equations effectively.