Solving the Given Linear Equation

We are asked to find the value of \(x\) in the linear equation:

\(\frac{5x}{3} - \frac{7}{2}\left(\frac{2x}{5} - \frac{1}{3}\right) = \frac{1}{3}\)

To solve this equation for \(x\), we need to simplify it by removing the parentheses and combining like terms. Let's break down the process into several steps.

Step-by-Step Method to Find the Value of x

Here is how we can solve this linear equation:

1. Simplify the expression inside the parentheses:
2. First, distribute the \(\frac{7}{2}\) to both terms inside the parentheses:
3. \(\frac{7}{2}\left(\frac{2x}{5} - \frac{1}{3}\right) = \left(\frac{7}{2} \times \frac{2x}{5}\right) - \left(\frac{7}{2} \times \frac{1}{3}\right)\)
4. \(= \frac{14x}{10} - \frac{7}{6}\)
5. Simplify the fraction \(\frac{14x}{10}\):
6. \(\frac{14x}{10} = \frac{7x}{5}\)
7. So the simplified expression is \(\frac{7x}{5} - \frac{7}{6}\).
8. Substitute the simplified expression back into the original equation:
9. The equation becomes:
10. \(\frac{5x}{3} - \left(\frac{7x}{5} - \frac{7}{6}\right) = \frac{1}{3}\)
11. Be careful with the minus sign before the parentheses. Distribute the minus sign:
12. \(\frac{5x}{3} - \frac{7x}{5} + \frac{7}{6} = \frac{1}{3}\)
13. Isolate terms with \(x\) on one side and constant terms on the other:
14. Subtract \(\frac{7}{6}\) from both sides of the equation:
15. \(\frac{5x}{3} - \frac{7x}{5} = \frac{1}{3} - \frac{7}{6}\)
16. Combine the terms with \(x\):
17. Find a common denominator for \(\frac{5x}{3}\) and \(\frac{7x}{5}\). The least common multiple (LCM) of 3 and 5 is 15.
18. \(\frac{5x}{3} = \frac{5x \times 5}{3 \times 5} = \frac{25x}{15}\)
19. \(\frac{7x}{5} = \frac{7x \times 3}{5 \times 3} = \frac{21x}{15}\)
20. Now combine them:
21. \(\frac{25x}{15} - \frac{21x}{15} = \frac{25x - 21x}{15} = \frac{4x}{15}\)
22. So the left side of the equation is \(\frac{4x}{15}\).
23. Combine the constant terms on the right side:
24. Find a common denominator for \(\frac{1}{3}\) and \(\frac{7}{6}\). The LCM of 3 and 6 is 6.
25. \(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)
26. Now combine them:
27. \(\frac{2}{6} - \frac{7}{6} = \frac{2 - 7}{6} = \frac{-5}{6}\)
28. So the right side of the equation is \(\frac{-5}{6}\).
29. Equate the simplified sides and solve for \(x\):
30. The equation is now:
31. \(\frac{4x}{15} = \frac{-5}{6}\)
32. To isolate \(x\), multiply both sides by 15:
33. \(4x = \frac{-5}{6} \times 15\)
34. \(4x = \frac{-75}{6}\)
35. Simplify the fraction \(\frac{-75}{6}\) by dividing the numerator and denominator by their greatest common divisor, which is 3:
36. \(\frac{-75 \div 3}{6 \div 3} = \frac{-25}{2}\)
37. So, \(4x = \frac{-25}{2}\).
38. Now, divide both sides by 4:
39. \(x = \frac{-25}{2} \div 4\)
40. \(x = \frac{-25}{2} \times \frac{1}{4}\)
41. \(x = \frac{-25 \times 1}{2 \times 4}\)
42. \(x = \frac{-25}{8}\)

Thus, the value of \(x\) that satisfies the given equation is \(\frac{-25}{8}\).

Revision Table: Steps to Solve Linear Equations

Additional Information on Solving Algebraic Equations

Solving linear equations involves isolating the variable using inverse operations. Key concepts include:

• Distributive Property: \(a(b+c) = ab + ac\). This is used to remove parentheses.
• Combining Like Terms: Terms with the same variable raised to the same power (or constant terms) can be added or subtracted.
• Properties of Equality:
  • Addition Property: Adding the same number to both sides of an equation keeps the equation balanced.
  • Subtraction Property: Subtracting the same number from both sides keeps the equation balanced.
  • Multiplication Property: Multiplying both sides by the same non-zero number keeps the equation balanced.
  • Division Property: Dividing both sides by the same non-zero number keeps the equation balanced.
• Least Common Multiple (LCM): Used to find a common denominator when adding or subtracting fractions. Multiplying the entire equation by the LCM of all denominators can clear the fractions, making the equation easier to solve. In step 6 above, instead of combining fractions first, we could have multiplied the entire equation \(\frac{5x}{3} - \frac{7x}{5} + \frac{7}{6} = \frac{1}{3}\) by the LCM of 3, 5, and 6 (which is 30) to clear the denominators.