Answer:
the value of \(x\) that satisfies the equation is \(\frac{9}{4}\).
Step-by-step explanation:
Sure, let's solve the equation step by step:
1. First, let's convert the mixed numbers to improper fractions:
\(1 \frac{2}{7} = \frac{9}{7}\)
\(2 \frac{1}{3} = \frac{7}{3}\)
2. Now, rewrite the equation:
\(\frac{9}{7} + \frac{7}{3}x + \frac{3}{14} + x = 9\)
3. Combine like terms:
\(\frac{9}{7} + \frac{3}{14} + \frac{7}{3}x + x = 9\)
4. Find a common denominator:
The common denominator is \(42\) (7 * 6 = 42, 14 * 3 = 42, 3 * 14 = 42).
5. Rewrite the equation with the common denominator:
\(\frac{9 \times 6}{7 \times 6} + \frac{3 \times 3}{14 \times 3} + \frac{7 \times 14}{3 \times 14}x + \frac{42}{42}x = 9\)
\( \frac{54}{42} + \frac{9}{42} + \frac{98}{42}x + \frac{42}{42}x = 9\)
6. Simplify:
\(\frac{54 + 9}{42} + \frac{98 + 42}{42}x = 9\)
\(\frac{63}{42} + \frac{140}{42}x = 9\)
7. Rewrite \(9\) as a fraction:
\(9 = \frac{378}{42}\)
8. Substitute back into the equation:
\(\frac{63}{42} + \frac{140}{42}x = \frac{378}{42}\)
9. Combine fractions:
\(\frac{63 + 140x}{42} = \frac{378}{42}\)
10. Now, equate the numerators:
\(63 + 140x = 378\)
11. Solve for \(x\):
\(140x = 378 - 63\)
\(140x = 315\)
\(x = \frac{315}{140}\)
\(x = \frac{45}{20}\)
\(x = \frac{9}{4}\)
So, the value of \(x\) that satisfies the equation is \(\frac{9}{4}\).
