Problem 37 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 3. $$ \frac{5}{6} k-7.5=7.5 $$

Step-by-Step Solution

Answer

The solution is \( k = 18 \).

Step 1: Add 7.5 to both sides

To isolate the term with the variable, first add 7.5 to both sides to eliminate the -7.5 on the left side of the equation. The equation becomes:\[ \frac{5}{6}k - 7.5 + 7.5 = 7.5 + 7.5 \]Simplifying, you get:\[ \frac{5}{6}k = 15 \]

Step 2: Multiply by the reciprocal

To isolate \( k \), multiply both sides by the reciprocal of \( \frac{5}{6} \), which is \( \frac{6}{5} \). This step allows us to cancel out the fraction:\[ \left(\frac{6}{5}\right) \times \frac{5}{6}k = 15 \times \frac{6}{5} \]Simplify:\[ k = 18 \]

Step 3: Check the solution

Substitute \( k = 18 \) back into the original equation to verify the solution:The original equation is:\[ \frac{5}{6}k - 7.5 = 7.5 \]Substitute \( k = 18 \):\[ \frac{5}{6}(18) - 7.5 \]Calculate \( \frac{5}{6} \times 18 = 15 \):\[ 15 - 7.5 = 7.5 \]Since both sides are equal, \( k = 18 \) is the correct solution.

Key Concepts

Reciprocal Operations

In the world of linear equations, reciprocal operations are your trusty tool to deal with fractions elegantly.
Reciprocal operations help us "unwind" fractions when solving equations, ensuring smoother navigation in the journey towards finding the variable.
If you have a fraction multiplying your variable, like in our example of \( \frac{5}{6}k = 15 \), identifying the reciprocal is crucial. What is a Reciprocal?
A reciprocal of a fraction is simply what you get when you flip the fraction. So, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \). You might remember this as "invert and multiply" from earlier math classes. Why Use Reciprocals?
Multiply both sides of the equation by this magical reciprocal to cancel out the fraction and isolate the variable.

• Ensure that any fraction becomes 1 when multiplied by its reciprocal because:\( \frac{5}{6} \times \frac{6}{5} = 1 \).
• This manipulation keeps the balance of the equation intact and leaves you with the variable all alone on one side.

By using reciprocal operations wisely, you can directly tackle any equation involving fractions with confidence.

Isolate the Variable

All roads in solving linear equations lead to isolating the variable. This is the strategy to find the unknown's value.
To "isolate the variable" means getting your variable all by itself on one side of the equation. This makes it easy to see what the variable equals, which is ultimately our goal. Let's walk through how we isolate the variable in our example. Step-by-Step Isolation

• Start by balancing the equation. If your variable is accompanied by constants, like "\( -7.5 \)" in the original equation \( \frac{5}{6}k - 7.5 = 7.5 \), you can first "undo" this constant by doing the opposite operation. Here, you'd add \( 7.5 \) to both sides, turning the equation into \( \frac{5}{6}k = 15 \).
• Eliminate any fractions or coefficients next. This often involves applying the reciprocal operation you learned. This step changes the state of the equation to show \( k = 18 \). Notice how this sequential unpeeling of mathematical layers leaves the variable splendidly isolated.

Isolating the variable is about untying the bonds between your variable and other numbers or operations. It is a process of simplicity revealing the unknown in your equation.

Checking Solutions

When solving equations, verification is key to ensuring accuracy. Once you have arrived at a solution, the best move involves checking your work by substituting back into the original equation. Here's why and how you should do it. Verification Process
Plugging your solution back into the equation checks the truthfulness of your findings. In our example, the solution \( k = 18 \) was obtained. Here's how we verify:

• Insert the value back into the original equation: Replace the variable with the solution found. Thus, the original \( \frac{5}{6}k - 7.5 = 7.5 \) becomes \( \frac{5}{6}(18) - 7.5 \).
• Calculate and compare both sides: Calculate to get \( 15 - 7.5 \), which equals \( 7.5 \). Notice both sides of the initial equation equal \( 7.5 \), confirming our solution is correct.

Make it a habit to check your solutions because it serves as both validation and a learning opportunity. This boost in confidence and understanding will help you tackle more complex problems efficiently.