Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-17\) solve each equation and check. $$ 2 y^{-1}=12 $$

Step-by-Step Solution

Answer

The solution is \(y = \frac{1}{6}\). Substitute to verify: the equation holds true.

Step 1: Understanding the Equation

The given equation is \(2y^{-1}=12\). Here, \(y^{-1}\) means the reciprocal of \(y\), which is \(\frac{1}{y}\). This means we can rewrite the equation as \(2 \cdot \frac{1}{y} = 12\).

Step 2: Isolate the Reciprocal of y

First, solve for \(\frac{1}{y}\) by dividing both sides of the equation by 2 to isolate the reciprocal. This gives us \(\frac{1}{y} = \frac{12}{2}\), simplifying to \(\frac{1}{y} = 6\).

Step 3: Solve for y

To solve for \(y\), take the reciprocal of both sides of the equation \(\frac{1}{y} = 6\), which gives \(y = \frac{1}{6}\).

Step 4: Check the Solution

Substitute \(y = \frac{1}{6}\) back into the original equation to verify. The equation becomes \(2 \cdot \left( \frac{1}{\frac{1}{6}} \right) = 2 \cdot 6 = 12\). The left-hand side equals 12, which matches the right-hand side, verifying that \(y = \frac{1}{6}\) is correct.

Key Concepts

Understanding Reciprocals

When solving equations such as \(2y^{-1} = 12\), understanding the concept of reciprocals is crucial. The reciprocal of a number is simply one divided by that number. In mathematical terms, the reciprocal of \(y\) is \(y^{-1}\), or \(\frac{1}{y}\). This means that when you see something like \(y^{-1}\) in an equation, you can think of it as \(\frac{1}{y}\).

• The operation of taking a reciprocal can be understood as flipping a fraction: if \(y = \, 6\), then \(y^{-1} = \frac{1}{6}\).
• This relationship is vital when working to solve equations involving reciprocals, especially when equations become complex.
• Always remember, every number except zero has a reciprocal, because you cannot divide by zero.

Techniques to Isolate Variables

Isolating variables is a fundamental technique in solving equations, and involves getting the variable by itself on one side of the equation. In the equation \(2 \cdot \frac{1}{y} = 12\), you should first focus on isolating \(\frac{1}{y}\).

• This is done by performing operations that will leave \(\frac{1}{y}\) alone on one side, such as dividing both sides by 2.
• After dividing, we get \(\frac{1}{y} = 6\). The operation applied must maintain the equation's balance, meaning whatever is done to one side, must also be done to the other.
• Once the reciprocal is isolated, solving for the variable \(y\) becomes straightforward with reciprocal actions, leading to \(y = \frac{1}{6}\).

This process may feel mechanical, but mastering it can lead to confident and efficient equation solving in mathematics.

Methodical Checking of Solutions

Checking solutions is an important step to ensure accuracy. After solving for \(y\), substituting the value back into the original equation \(2y^{-1} = 12\) helps verify the solution.

• Substitute back: if \(y = \frac{1}{6}\), then \(y^{-1} = 6\). Replacing into the original equation gives \(2 \cdot 6 = 12\).
• Evaluate both sides: Confirm that both the left-hand side and right-hand side of the equation are equal. In this case, both equal 12.
• If both sides match, the solution is correct. If not, re-evaluate your steps as there might be errors.

Checking your solutions not only verifies correctness but also helps reinforce understanding of the mathematical processes involved.