Solve for x.
We are asked to solve two linear equations for the variable \(x\).
Equation (a) is \(x + 3 \frac{1}{2} = 4 \frac{1}{3}\) and equation (b) is \(x - 2 \frac{1}{2} = \frac{3}{4}\).
A mixed number represents the sum of its integer part and its fractional part. For example, \(a \frac{b}{c} = a + \frac{b}{c} = \frac{ac+b}{c}\).
Our strategy is to convert all mixed numbers into improper fractions, isolate \(x\) by performing the necessary arithmetic operations, and then simplify the results back into a mixed number or proper fraction as needed.
For equation (a), we have:
\[x + 3 \frac{1}{2} = 4 \frac{1}{3}\]
First, convert the mixed numbers to improper fractions:
\[3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\]
\[4 \frac{1}{3} = 4 + \frac{1}{3} = \frac{12}{3} + \frac{1}{3} = \frac{13}{3}\]
Now substitute these into the equation:
\[x + \frac{7}{2} = \frac{13}{3}\]
To isolate \(x\), subtract \(\frac{7}{2}\) from both sides:
\[x = \frac{13}{3} - \frac{7}{2}\]
To subtract these, find a common denominator, which is 6:
\[x = \frac{13 \times 2}{6} - \frac{7 \times 3}{6} = \frac{26}{6} - \frac{21}{6} = \frac{5}{6}\]
The solution for equation (a) is \(x = \frac{5}{6}\).
For equation (b), we have:
\[x - 2 \frac{1}{2} = \frac{3}{4}\]
First, convert the mixed number to an improper fraction:
\[2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}\]
Substitute this into the equation:
\[x - \frac{5}{2} = \frac{3}{4}\]
To isolate \(x\), add \(\frac{5}{2}\) to both sides:
\[x = \frac{3}{4} + \frac{5}{2}\]
Find a common denominator, which is 4:
\[x = \frac{3}{4} + \frac{5 \times 2}{4} = \frac{3}{4} + \frac{10}{4} = \frac{13}{4}\]
Convert the improper fraction back to a mixed number:
\[\frac{13}{4} = 3 \frac{1}{4}\]
The solution for equation (b) is \(x = 3 \frac{1}{4}\).
We have solved both equations successfully. For equation (a), \(x = \frac{5}{6}\). For equation (b), \(x = 3 \frac{1}{4}\). Both solutions satisfy the given equations and are in the required format.
Understanding how to handle equations with fractions and mixed numbers is essential for real-life tasks like cooking or carpentry. For example, if a recipe calls for \(3 \frac{1}{2}\) cups of flour, but you only have \(4 \frac{1}{3}\) cups available, you can use these algebraic skills to calculate exactly how much more flour you would need or how much you have leftover. By converting these measurements into improper fractions, you can perform precise arithmetic to ensure your projects and recipes come out perfectly every time.