Question

Solve equation for the unknown quantity. Check your answers. \(2 \frac{1}{3}+x=5 \frac{5}{6}\)

Short answer

Question: Solve for x in the equation \(2 \frac{1}{3}+x=5 \frac{5}{6}\). Answer: \(x = \frac{7}{2}\).

Step-by-step solution

Convert mixed numbers to improper fractions

To convert the mixed numbers into improper fractions, we multiply the whole number by the denominator, add the numerator, and put the result over the denominator. \(2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}\) \(5 \frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{35}{6}\) Now, the equation becomes: \(\frac{7}{3} + x = \frac{35}{6}\)

Find a common denominator.

To solve for x, we must first find a common denominator for the fractions. In this case, the lowest common denominator (LCD) is 6. Therefore, we will multiply both numerator and denominator of \(\frac{7}{3}\) by 2 to get the same denominator. \(\frac{7}{3} \times \frac{2}{2} = \frac{14}{6}\) So, the equation becomes: \(\frac{14}{6} + x = \frac{35}{6}\)

Subtract \(\frac{14}{6}\) from both sides.

In order to isolate x, we need to move the \(\frac{14}{6}\) term to the other side of the equation. We do this by subtracting \(\frac{14}{6}\) from both sides. \(\frac{14}{6} + x - \frac{14}{6} = \frac{35}{6} - \frac{14}{6}\) \(x = \frac{21}{6}\)

Simplify the fraction

Simplify the fraction \(\frac{21}{6}\) by dividing the numerator and denominator by their greatest common divisor (GCD). The GCD in this case is 3. \(x = \frac{21}{6} \div \frac{3}{3} = \frac{7}{2}\) Therefore, the value of x is \(\frac{7}{2}\).

Check the solution

To check the solution, plug the value of x back into the original equation and see if the equation holds true. \(2 \frac{1}{3}+\frac{7}{2}=5 \frac{5}{6}\) Converting the mixed numbers to improper fractions: \(\frac{7}{3}+\frac{7}{2}=\frac{35}{6}\) Convert both fractions to the same denominator (LCD = 6): \(\frac{14}{6}+\frac{21}{6}=\frac{35}{6}\) Combine the fractions: \(\frac{35}{6}=\frac{35}{6}\) The equation balances, which verifies that the solution is correct: \(x=\frac{7}{2}\)