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Chapter 7: Problem 15

Solve each equation: $$ \frac{17}{3} t=-2 $$

Short Answer

Expert verified
\( t = \frac{-6}{17} \).

Step by step solution

01

Understand the Equation

The equation given is \( \frac{17}{3} t = -2 \). The goal is to solve for \( t \), which means we need to isolate \( t \) on one side of the equation. To do this, we need to eliminate the fraction.
02

Clear the Fraction

To clear the fraction \( \frac{17}{3} \), multiply both sides of the equation by its reciprocal, \( \frac{3}{17} \). This will help isolate \( t \).
03

Simplify Both Sides

Multiply the left side: \( \frac{3}{17} \times \frac{17}{3} t = t \). For the right side, \( \frac{3}{17} \times -2 = \frac{-6}{17} \). After multiplying both sides by \( \frac{3}{17} \), the equation becomes \( t = \frac{-6}{17} \).

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Key Concepts

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

Fraction Multiplication
When dealing with equations that contain fractions, like \( \frac{17}{3} t = -2 \), it's important to have a solid understanding of fraction multiplication. Multiplying fractions involves multiplying the numerators (top numbers) together and the denominators (bottom numbers) together. For example, to multiply \( \frac{3}{17} \) by \( \frac{17}{3} \):
  • Multiply the numerators: \( 3 \times 17 = 51 \)
  • Multiply the denominators: \( 17 \times 3 = 51 \)
Thus, \( \frac{3}{17} \times \frac{17}{3} = \frac{51}{51} = 1 \). This is helpful because multiplying by the reciprocal cancels out the fraction, simplifying your equation. Remember, finding and using the reciprocal (flipping the fraction) helps to eliminate fractions when solving equations, making the process more straightforward.
Isolation of Variable
Isolating the variable is a key step in solving equations, particularly linear equations like \( \frac{17}{3} t = -2 \). To isolate the variable means to get \( t \) by itself on one side of the equation. Here's how it's done:
  • Identify the term with the variable: in this case, it's \( \frac{17}{3} t \).
  • Since \( \frac{17}{3} \) is a coefficient of \( t \), multiply both sides of the equation by its reciprocal, \( \frac{3}{17} \), to cancel it out.
  • By multiplying \( \frac{3}{17} \times \frac{17}{3} \), you simplify it to 1, effectively leaving \( t \) isolated.
After isolating the variable, you'll determine the value of \( t \): \( t = \frac{-6}{17} \). This shows why understanding how to manipulate fractions is critical when isolating variables.
Solving Equations
To solve equations like \( \frac{17}{3} t = -2 \), it's important to go through a systematic process. Here鈥檚 a simple guide:
  • Start by understanding the equation and what it's asking you to find.
  • Identify any fractions involved and clear them by using the reciprocal, as multiplied against both sides of the equation.
  • Simplify the equation, ensuring the variable is isolated on one side.
  • Reexamine each step to confirm the accuracy of your simplification and calculations.
In our example, after clearing the fraction and isolating \( t \), we found \( t = \frac{-6}{17} \). Thus, solving equations not only involves computation but also logic and verification to ensure each step leads closer to a meaningful solution.

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Most popular questions from this chapter

Solve each equation: $$ -7 x+1=-6 $$

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If \(a\) is inversely proportional to \(b\) state which of the following are true and which are false: (a) \(a\) multiplied by \(b\) is a constant (b) \(a\) divided by \(b\) is a constant (c) \(a^{2}\) is inversely proportional to \(b^{2}\)

Show that \(3-2 t-t^{2}=-(t+3)(t-1)\).
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