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Chapter 5: Problem 23

Solve equation for the unknown quantity. Check your answers. \(x-3 \frac{1}{2}=6 \frac{3}{4}\)

Short Answer

Expert verified
Question: Solve for x in the equation \(x - 3\frac{1}{2} = 6\frac{3}{4}\). Answer: \(x = \frac{41}{4}\)

Step by step solution

01

Convert mixed numbers to improper fractions

First, let's convert the mixed numbers into improper fractions. To do this, we will multiply the whole number by the denominator and then add the numerator. The resulting fraction will have the same denominator. For \(3\frac{1}{2}\): \(3 \times 2 + 1 = 7\) So, the improper fraction is \(\frac{7}{2}\) For \(6\frac{3}{4}\): \(6 \times 4 + 3 = 27\) So, the improper fraction is \(\frac{27}{4}\) Now, the equation becomes: \(x - \frac{7}{2} = \frac{27}{4}\)
02

Isolate x

To isolate x, we will add \(\frac{7}{2}\) to both sides of the equation. \(x - \frac{7}{2} + \frac{7}{2} = \frac{27}{4} + \frac{7}{2}\) Now, we can simplify the right side by finding a common denominator. In this case, the common denominator is 4. So, we will convert the \(\frac{7}{2}\) into a fraction with a denominator of 4. \(\frac{7}{2} \times \frac{2}{2} = \frac{14}{4}\) Now, the equation becomes: \(x = \frac{27}{4} + \frac{14}{4}\)
03

Solve for x

Now, we can simply add the fractions on the right side of the equation: \(x = \frac{27 + 14}{4}\) \(x = \frac{41}{4}\) So, our answer is \(x = \frac{41}{4}\).
04

Check the answer

Now, let's check our answer by plugging it back into the original equation: \(x - 3\frac{1}{2} = 6\frac{3}{4}\) \(\frac{41}{4} - 3\frac{1}{2} = 6\frac{3}{4}\) Convert the mixed numbers to improper fractions: \(\frac{41}{4} - \frac{7}{2} = \frac{27}{4}\) Find a common denominator to subtract the fractions: \(\frac{41}{4} - \frac{14}{4} = \frac{27}{4}\) Simplify: \(\frac{27}{4} = \frac{27}{4}\) Our answer checks out, so our solution of \(x = \frac{41}{4}\) is correct.

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