Problem 30 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{y}{-3}=-4 $$

Step-by-Step Solution

Answer

Answer: The value of y that satisfies the given conditional equation is 12.

Step 1: Identify the equation to solve

We are given the equation: $$ \frac{y}{-3}=-4 $$ Our task is to solve for the variable y.

Step 2: Get rid of the fraction

To get rid of the fraction, we can multiply both sides of the equation by the denominator of the fraction, which is -3. So, we have: $$ -3\left(\frac{y}{-3}\right)=-4(-3) $$

Step 3: Simplify the equation

Now, we need to simplify the equation: $$ \frac{-3y}{-3} = 12 $$ The -3 in the numerator and the denominator cancel each other out, leaving us with: $$ y = 12 $$

Step 4: Write the solution

The solution to the equation is: $$ y = 12 $$ So, the value of y that satisfies the given conditional equation is 12.

Key Concepts

Solving Conditional Equations

When solving equations, our main aim is to find the value of the variable that turns the equation into a true statement. Conditional equations mean that certain values for the variable make the statement true, and other values do not. In this exercise, we must find the value of \( y \) that satisfies the equation \( \frac{y}{-3} = -4 \).
To do this, we often manipulate or rearrange the equation until we isolate the variable by itself on one side of the equation.
This step-by-step manipulation brings us closer to understanding what specific value of the variable makes the equation work. Remember, the goal is for the left and right sides of the equation to equal one another.

Fractions in Equations

Fractions can sometimes make equations look complicated, but they don't have to be intimidating! In our equation \( \frac{y}{-3} = -4 \), the fraction is \( \frac{y}{-3} \). A fraction consists of a numerator and a denominator. Here, the denominator is \(-3\).

To "get rid" of the fraction, we use a simple trick: multiply both sides of the equation by the denominator. This ensures that the equation remains balanced and the fraction disappears.

• By multiplying both sides by \(-3\), we perform the same operation on each side, thus respecting the equation's balance.
• This method works because multiplying by \(-3\) effectively "cancels out" the division by \(-3\), leaving the numerator \( y \) alone.

In the end, it's just a series of steps that simplify the equation.

Simplifying Equations

Once we eliminate the fraction by multiplying through by its denominator, the equation \( -3\left(\frac{y}{-3}\right) = -4(-3) \) becomes simpler. We then proceed to carefully simplify both sides.
Here's how:

• On the left side, multiplying \(-3\) by \( \frac{y}{-3} \) simplifies to \( y \), as the \(-3\) in the numerator and denominator cancels out.
• On the right side, multiplying \(-4\) by \(-3\) gives \( 12 \), because two negatives make a positive.

Finally, we can clearly see that our simplified equation is \( y = 12 \). This simplification process confirms our solution and allows us to double-check that no mistakes were made along the way.
By breaking down each part of the equation and logically simplifying it, we ensure our solution is both accurate and reliable.