Question

Solve equation for the unknown quantity. Check your answers. \(x+5 \frac{1}{4}=-7 \frac{1}{3}\)

Short answer

Answer: The value of x is \(\frac{-143}{12}\).

Step-by-step solution

Convert mixed numbers to improper fractions

In order to work with the equation more easily, first let's convert the mixed numbers (5 1/4 and -7 1/3) to improper fractions. To do this, multiply the whole number by the denominator and then add the numerator. The denominator stays the same. For 5 1/4: (5 × 4) + 1 = 21, so the improper fraction is 21/4. For -7 1/3: (-7 × 3) + 1 = -20, so the improper fraction is -20/3. Now the equation is: \(x + \frac{21}{4} = \frac{-20}{3}\)

Subtract the fraction from both sides of the equation

In order to isolate the variable x, subtract the 21/4 fraction from both sides of the equation. \(x = \frac{-20}{3} - \frac{21}{4}\)

Find a common denominator to subtract the fractions

To subtract the fractions, we need a common denominator. The least common multiple (LCM) of 3 and 4 is 12. We will convert the fractions to have a denominator of 12, by multiplying the numerator and the denominator with the appropriate value. For -20/3: \(- \frac{20}{3} \cdot \frac{4}{4} = \frac{-80}{12}\) For 21/4: \(\frac{21}{4} \cdot \frac{3}{3} = \frac{63}{12}\) Now the equation is: \(x = \frac{-80}{12} - \frac{63}{12}\)

Subtract the fractions

Now we can subtract the fractions easily since they have the same denominator: \(x = \frac{-80 - 63}{12}\) Add the numerators: \(x = \frac{-143}{12}\)

Check the solution

To check the solution, we will plug the value of x (-143/12) back into the original equation and see if both sides are equal. Original equation: \(x + 5 \frac{1}{4} = -7 \frac{1}{3}\) Plugging in the value of x: \(\frac{-143}{12} + 5 \frac{1}{4} = -7 \frac{1}{3}\) We already converted the mixed numbers to improper fractions in step 1, so the equation then becomes: \(\frac{-143}{12} + \frac{21}{4} = \frac{-20}{3}\) Following step 2 to step 4, we find that both sides are equal, so the solution is correct. Therefore, the value of x is \(x = \frac{-143}{12}\).