Solve for u u+3/4=1/8

Solution:

Step 1:

Isolate the variable $u$ by eliminating $\frac{3}{4}$ from both sides of the equation: $u = \frac{1}{8} - \frac{3}{4}$.

Step 2:

Convert $-\frac{3}{4}$ to have the same denominator as $\frac{1}{8}$ by multiplying by $\frac{2}{2}$: $u = \frac{1}{8} - \frac{3}{4} \cdot \frac{2}{2}$.

Step 3:

Ensure both fractions have a denominator of $8$ by finding equivalent fractions.

Step 3.1:

Multiply the numerator and denominator of $\frac{3}{4}$ by $2$: $u = \frac{1}{8} - \frac{3 \cdot 2}{4 \cdot 2}$.

Step 3.2:

Confirm the common denominator of $8$: $u = \frac{1}{8} - \frac{3 \cdot 2}{8}$.

Step 4:

Combine the fractions over the same denominator: $u = \frac{1 - 3 \cdot 2}{8}$.

Step 5:

Simplify the numerator by performing the arithmetic operations.

Step 5.1:

Calculate $3$ times $2$: $u = \frac{1 - 6}{8}$.

Step 5.2:

Subtract $6$ from $1$: $u = \frac{-5}{8}$.

Step 6:

Position the negative sign in front of the fraction: $u = -\frac{5}{8}$.

Step 7:

Express the solution in various forms.

Exact Form: $u = -\frac{5}{8}$

Decimal Form: $u = -0.625$

Knowledge Notes:

The problem involves solving a simple linear equation for the variable $u$. The steps taken to solve the equation include:

1. Isolating the Variable: The goal is to get $u$ by itself on one side of the equation. This often involves undoing addition or subtraction on the same side as the variable.

2. Finding a Common Denominator: When dealing with fractions, it's essential to have a common denominator to combine terms. In this case, the common denominator is $8$.

3. Equivalent Fractions: Multiplying the numerator and denominator of a fraction by the same number creates an equivalent fraction, which is a fraction that has the same value as the original.

4. Combining Fractions: Once the fractions have the same denominator, their numerators can be combined (added or subtracted) while keeping the common denominator.

5. Simplifying the Expression: After combining the fractions, the numerator may need to be simplified by performing the addition or subtraction.

6. Negative Fractions: A negative sign in front of a fraction applies to the entire fraction, not just the numerator or the denominator.

7. Multiple Forms of the Solution: The solution to an equation can be presented in different forms, such as an exact fraction or a decimal.

Understanding these concepts is crucial for solving linear equations and working with fractions in algebra.