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Chapter 2: Problem 27

Solve and check. Label any contradictions or identities. $$\frac{3}{5} t-1=8$$

Short Answer

Expert verified
t = 15

Step by step solution

01

- Isolate the variable term

Add 1 to both sides of the equation to get: \[\frac{3}{5}t - 1 + 1 = 8 + 1\] This simplifies to: \[\frac{3}{5}t = 9\]
02

- Solve for the variable

To isolate \(t\), multiply both sides of the equation by the reciprocal of \(\frac{3}{5}\): \[\left(\frac{5}{3}\right) \cdot \frac{3}{5}t = 9 \cdot \left(\frac{5}{3}\right)\] This simplifies to: \[t = 15\]
03

- Check the solution

Substitute \(t = 15\) back into the original equation to check the solution: \[\frac{3}{5}(15) - 1 = 8\] Calculate the left-hand side: \[9 - 1 = 8\] Since both sides of the equation are equal, \(t = 15\) is a correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the variable
To solve a linear equation, the first key step is isolating the variable. This means you want to get the variable by itself on one side of the equation. Let's start with the given equation: \[\frac{3}{5}t - 1 = 8\]Since we want to isolate the variable term \(\frac{3}{5}t\), we need to remove the constant \(-1\) on the left side. To do this, we add 1 to both sides: \[\frac{3}{5}t - 1 + 1 = 8 + 1\]Simplifying this gives: \[\frac{3}{5}t = 9\]Now, the variable term \(\frac{3}{5}t\) is isolated. By performing these operations on both sides of the equation, we maintain equality and move closer to finding the value of the variable.
Reciprocal
Once the variable term is isolated, the next step involves using the reciprocal to solve for the variable. The reciprocal is the flipped version of a fraction. For example, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). To isolate t, we will multiply both sides of the equation \[\frac{3}{5}t = 9\]by the reciprocal of \(\frac{3}{5}\):\[\frac{5}{3} \times \frac{3}{5}t = 9 \times \frac{5}{3} \]This simplifies to:\[t = 15\]We used the reciprocal to cancel out the fraction. Multiplying \(\frac{3}{5}\) by \(\frac{5}{3}\) equals 1, leaving us with t by itself, hence solving the equation.
Checking solutions
Finally, it's crucial to check the solution to ensure it is correct. Substituting the value we found for t back into the original equation will help with this. We substitute t = 15 back into the equation:\[\frac{3}{5}(15) - 1 = 8\]Calculate the left-hand side step by step: \[\frac{3}{5}(15) = 9\]Then: \[9 - 1 = 8\]We see that both sides of the equation are equal, confirming that our solution \t = 15\ is correct. This process helps verify the accuracy and validity of our solution, ensuring no mistakes were made during calculations.

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