Solve for k 1/2k+2=5+5/6k

Solution:

Step 1

Merge the terms involving $k$ and constants separately: $\frac{1}{2}k + 2 = 5 + \frac{5}{6}k$

Step 2

Rewrite the equation with $k$ terms together: $\frac{1}{2}k + 2 = 5 + \frac{5}{6}k$

Step 3

Isolate the terms with $k$ on one side of the equation.

Step 3.1

Subtract $\frac{5}{6}k$ from both sides: $\frac{1}{2}k + 2 - \frac{5}{6}k = 5$

Step 3.2

Convert $\frac{1}{2}k$ to have a denominator of 6 by multiplying by $\frac{3}{3}$: $\frac{1}{2}k \cdot \frac{3}{3} - \frac{5}{6}k + 2 = 5$

Step 3.3

Ensure all terms have a common denominator.

Step 3.3.1

Multiply $\frac{1}{2}k$ by $\frac{3}{3}$: $\frac{3k}{6} - \frac{5}{6}k + 2 = 5$

Step 3.3.2

Multiply 2 by $\frac{6}{6}$: $\frac{3k}{6} - \frac{5}{6}k + \frac{12}{6} = 5$

Step 3.4

Combine like terms over the common denominator: $\frac{3k - 5k}{6} + \frac{12}{6} = 5$

Step 3.5

Simplify the numerator by combining $k$ terms.

Step 3.5.1

Combine $3k$ and $-5k$: $\frac{-2k}{6} + \frac{12}{6} = 5$

Step 3.6

Reduce the fraction by cancelling common factors.

Step 3.6.1

Factor out a 2 from the numerator and denominator: $\frac{-k}{3} + \frac{12}{6} = 5$

Step 3.6.2

Simplify the constant term: $\frac{-k}{3} + 2 = 5$

Step 4

Move constant terms to the opposite side of the equation.

Step 4.1

Subtract 2 from both sides: $\frac{-k}{3} = 5 - 2$

Step 4.2

Calculate the difference: $\frac{-k}{3} = 3$

Step 5

Eliminate the fraction by multiplying both sides by -3: $-3 \cdot \frac{-k}{3} = -3 \cdot 3$

Step 6

Simplify both sides of the equation.

Step 6.1

Simplify the left side by cancelling out the 3s: $k = -3 \cdot 3$

Step 6.2

Multiply the constants on the right side: $k = -9$

Knowledge Notes:

To solve an equation involving fractions and a variable, the following knowledge points are relevant:

1. Combining Like Terms: When solving equations, combine terms with the variable and constants separately to simplify the equation.

2. Common Denominator: To combine fractions, they must have a common denominator. Multiply by a form of 1 (such as $\frac{3}{3}$) to achieve this without changing the value.

3. Isolating the Variable: Move all terms with the variable to one side and constants to the other side to isolate the variable.

4. Simplifying Fractions: Cancel common factors in the numerator and denominator to simplify fractions.

5. Inverse Operations: Use inverse operations to eliminate coefficients or fractions from the variable. For example, multiply both sides by the reciprocal to clear a fraction.

6. Solving Linear Equations: The goal is to isolate the variable on one side to find its value. Perform the same operation on both sides of the equation to maintain equality.

7. Multiplication and Division: These are inverse operations used to solve for a variable. Multiplication by a negative number changes the sign of the product.