Solve for z 1 3/4+z=2 2/3

Solution:

Step 1: Simplify the equation's left-hand side.

• Step 1.1: Transform $1 \frac{3}{4}$ into an improper fraction.

  • Step 1.1.1: Decompose the mixed number into its integer and fractional parts: $1 + \frac{3}{4} + z = 2 \frac{2}{3}$.

  • Step 1.1.2: Combine the integer and the fraction.

    • Step 1.1.2.1: Express the integer 1 as a fraction: $\frac{4}{4} + \frac{3}{4} + z = 2 \frac{2}{3}$.

    • Step 1.1.2.2: Add the fractions with a common denominator: $\frac{4 + 3}{4} + z = 2 \frac{2}{3}$.

    • Step 1.1.2.3: Perform the addition: $\frac{7}{4} + z = 2 \frac{2}{3}$.

Step 2: Simplify the equation's right-hand side.

• Step 2.1: Convert $2 \frac{2}{3}$ into an improper fraction.

  • Step 2.1.1: Break down the mixed number: $\frac{7}{4} + z = 2 + \frac{2}{3}$.

  • Step 2.1.2: Sum the integer and the fraction.

    • Step 2.1.2.1: Represent the integer 2 as a fraction: $\frac{7}{4} + z = 2 \cdot \frac{3}{3} + \frac{2}{3}$.

    • Step 2.1.2.2: Merge the integer and fraction: $\frac{7}{4} + z = \frac{2 \cdot 3}{3} + \frac{2}{3}$.

    • Step 2.1.2.3: Add the fractions with a common denominator: $\frac{7}{4} + z = \frac{2 \cdot 3 + 2}{3}$.

    • Step 2.1.2.4: Simplify the numerator.

      • Step 2.1.2.4.1: Multiply 2 by 3: $\frac{7}{4} + z = \frac{6 + 2}{3}$.

      • Step 2.1.2.4.2: Complete the addition: $\frac{7}{4} + z = \frac{8}{3}$.

Step 3: Isolate the variable $z$.

• Step 3.1: Subtract $\frac{7}{4}$ from both sides: $z = \frac{8}{3} - \frac{7}{4}$.

• Step 3.2: Find a common denominator for $\frac{8}{3}$.

• Step 3.3: Find a common denominator for $-\frac{7}{4}$.

• Step 3.4: Express both fractions with a common denominator of $12$.

  • Step 3.4.1: Scale $\frac{8}{3}$ to have the denominator 12: $z = \frac{8 \cdot 4}{3 \cdot 4} - \frac{7}{4} \cdot \frac{3}{3}$.

  • Step 3.4.2: Multiply 3 by 4: $z = \frac{8 \cdot 4}{12} - \frac{7}{4} \cdot \frac{3}{3}$.

  • Step 3.4.3: Scale $\frac{7}{4}$ to have the denominator 12: $z = \frac{8 \cdot 4}{12} - \frac{7 \cdot 3}{4 \cdot 3}$.

  • Step 3.4.4: Multiply 4 by 3: $z = \frac{8 \cdot 4}{12} - \frac{7 \cdot 3}{12}$.

• Step 3.5: Combine the numerators over the common denominator: $z = \frac{8 \cdot 4 - 7 \cdot 3}{12}$.

• Step 3.6: Simplify the numerator.

  • Step 3.6.1: Multiply 8 by 4: $z = \frac{32 - 7 \cdot 3}{12}$.

  • Step 3.6.2: Multiply -7 by 3: $z = \frac{32 - 21}{12}$.

  • Step 3.6.3: Subtract 21 from 32: $z = \frac{11}{12}$.

Step 4: Present the result in various forms.

• Exact Form: $z = \frac{11}{12}$
• Decimal Form: $z \approx 0.9167$

Knowledge Notes:

To solve the equation $1 \frac{3}{4} + z = 2 \frac{2}{3}$, we need to follow a systematic process:

1. Conversion of Mixed Numbers to Improper Fractions: Mixed numbers are converted to improper fractions to simplify the addition or subtraction of fractions. A mixed number is the sum of its whole number part and its fractional part.

2. Finding a Common Denominator: When adding or subtracting fractions, it's essential to have a common denominator. This allows us to combine the numerators directly.

3. Combining Like Terms: After converting to a common denominator, we combine like terms by adding or subtracting the numerators while keeping the denominator the same.

4. Isolating the Variable: To solve for the variable $z$, we need to isolate it on one side of the equation. This is done by performing the same operation on both sides of the equation to maintain equality.

5. Simplification: The final step involves simplifying the fraction by performing the arithmetic operations in the numerator and then reducing the fraction to its simplest form if possible.

6. Multiple Forms of the Result: The solution can be presented in its exact form as a fraction or converted to a decimal for a more intuitive understanding.