Solve for k (k+1)/2+(k+3)/4=1/2

Solution:

Step:1

Transform $\frac{k + 1}{2} + \frac{k + 3}{4}$ into a single fraction.

Step:1.1

Convert $\frac{k + 1}{2}$ to have a denominator of $4$ by multiplying by $\frac{2}{2}$. $\frac{k + 1}{2} \cdot \frac{2}{2} + \frac{k + 3}{4} = \frac{1}{2}$

Step:1.2

Adjust each term to a denominator of $4$.

Step:1.2.1

Scale $\frac{k + 1}{2}$ by $\frac{2}{2}$. $\frac{(k + 1) \cdot 2}{2 \cdot 2} + \frac{k + 3}{4} = \frac{1}{2}$

Step:1.2.2

Multiply $2$ by itself. $\frac{(k + 1) \cdot 2}{4} + \frac{k + 3}{4} = \frac{1}{2}$

Step:1.3

Merge the numerators over the shared denominator. $\frac{(k + 1) \cdot 2 + k + 3}{4} = \frac{1}{2}$

Step:1.4

Streamline the numerator.

Step:1.4.1

Employ the distributive law. $\frac{k \cdot 2 + 1 \cdot 2 + k + 3}{4} = \frac{1}{2}$

Step:1.4.2

Reposition $2$ before $k$. $\frac{2 \cdot k + 2 + k + 3}{4} = \frac{1}{2}$

Step:1.4.3

Execute the multiplication of $2$ by $1$. $\frac{2 \cdot k + 2 + k + 3}{4} = \frac{1}{2}$

Step:1.4.4

Combine $2k$ with $k$. $\frac{3k + 5}{4} = \frac{1}{2}$

Step:2

Cross-multiply by $4$. $\frac{3k + 5}{4} \cdot 4 = \frac{1}{2} \cdot 4$

Step:3

Condense the equation.

Step:3.1

Refine the left-hand side.

Step:3.1.1

Eliminate the $4$.

Step:3.1.1.1

Remove the $4$. $\frac{3k + 5}{\cancel{4}} \cdot \cancel{4} = \frac{1}{2} \cdot 4$

Step:3.1.1.2

Express the simplified form. $3k + 5 = 2$

Step:3.2

Clarify the right-hand side.

Step:3.2.1

Extract the $2$.

Step:3.2.1.1

Isolate $2$ from $4$. $3k + 5 = \frac{1}{2} \cdot (2 \cdot 2)$

Step:3.2.1.2

Cancel out the $2$. $3k + 5 = 1 \cdot (2)$

Step:3.2.1.3

Rephrase the expression. $3k + 5 = 2$

Step:4

Determine the value of $k$.

Step:4.1

Shift constants to the opposite side.

Step:4.1.1

Subtract $5$ on both sides. $3k = 2 - 5$

Step:4.1.2

Compute $2 - 5$. $3k = -3$

Step:4.2

Divide the equation by $3$.

Step:4.2.1

Divide $3k$ and $-3$ by $3$. $\frac{3k}{3} = \frac{-3}{3}$

Step:4.2.2

Simplify the left-hand side.

Step:4.2.2.1

Remove the $3$. $\frac{\cancel{3}k}{\cancel{3}} = \frac{-3}{3}$

Step:4.2.3

Simplify the right-hand side.

Step:4.2.3.1

Divide $-3$ by $3$. $k = -1$

Knowledge Notes:

The problem involves solving a linear equation with fractions. The key knowledge points include:

1. Common Denominator: To combine fractions, they must have the same denominator. Multiplying by a form of one (e.g., $\frac{2}{2}$) that does not change the value of the fraction is used to achieve this.

2. Distributive Property: This property is used to simplify expressions, such as $2(k + 1)$, which becomes $2k + 2$.

3. Cross-Multiplication: When two fractions are set equal to each other, you can cross-multiply to eliminate the denominators.

4. Simplifying Equations: This involves canceling out common factors on both sides of the equation to isolate the variable.

5. Solving Linear Equations: The goal is to isolate the variable (in this case, $k$) by performing inverse operations (addition/subtraction, multiplication/division) on both sides of the equation.

Understanding these concepts is crucial for solving linear equations and working with algebraic expressions involving fractions.