Solve for x 1/6(x+1/5)=-17/15

Solution:

Step:1

Eliminate the fraction on the left side by multiplying the entire equation by $6$.

$6 \cdot \left( \frac{1}{6}(x + \frac{1}{5}) \right) = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2

Begin simplification of both sides of the equation.

Step:2.1

Start with the left side.

Step:2.1.1

Expand $6 \cdot \left( \frac{1}{6}(x + \frac{1}{5}) \right)$.

Step:2.1.1.1

Distribute $6$ across the terms inside the parentheses.

$6 \cdot \left( \frac{1}{6}x + \frac{1}{6} \cdot \frac{1}{5} \right) = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.2

Combine $6$ and $x$ divided by $6$.

$6 \cdot \left( \frac{x}{6} + \frac{1}{6} \cdot \frac{1}{5} \right) = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.3

Perform the multiplication of $\frac{1}{6} \cdot \frac{1}{5}$.

Step:2.1.1.3.1

Multiply $\frac{1}{6}$ by $\frac{1}{5}$.

$6 \cdot \left( \frac{x}{6} + \frac{1}{6 \cdot 5} \right) = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.3.2

Calculate $6$ times $5$.

$6 \cdot \left( \frac{x}{6} + \frac{1}{30} \right) = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.4

Apply the distributive property to the left side.

$6 \cdot \frac{x}{6} + 6 \cdot \frac{1}{30} = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.5

Eliminate the common factor of $6$ on the left side.

Step:2.1.1.5.1

Remove the common factor.

$x + 6 \cdot \frac{1}{30} = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.5.2

Rewrite the simplified left side.

$x + \frac{6}{30} = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.1.1.6

Simplify the fraction on the left side.

Step:2.1.1.6.1

Extract $6$ from $30$.

$x + \frac{1}{5} = 6 \cdot \left( -\frac{17}{15} \right)$

Step:2.2

Now, simplify the right side of the equation.

Step:2.2.1

Work on simplifying $6 \cdot \left( -\frac{17}{15} \right)$.

Step:2.2.1.1

Remove the common factors.

Step:2.2.1.1.1

Move the negative sign to the numerator.

$x + \frac{1}{5} = 6 \cdot \left( \frac{-17}{15} \right)$

Step:2.2.1.1.2

Extract $3$ from $6$.

$x + \frac{1}{5} = 2 \cdot 3 \cdot \frac{-17}{15}$

Step:2.2.1.1.3

Extract $3$ from $15$.

$x + \frac{1}{5} = 2 \cdot \frac{-17}{5}$

Step:2.2.1.1.4

Cancel out the common factor.

$x + \frac{1}{5} = 2 \cdot \frac{-17}{5}$

Step:2.2.1.2

Combine $2$ with $\frac{-17}{5}$.

$x + \frac{1}{5} = \frac{-34}{5}$

Step:2.2.1.3

Simplify the right side.

Step:2.2.1.3.1

Multiply $2$ by $-17$.

$x + \frac{1}{5} = -\frac{34}{5}$

Step:3

Isolate $x$ by moving terms without $x$ to the other side.

Step:3.1

Subtract $\frac{1}{5}$ from both sides.

$x = -\frac{34}{5} - \frac{1}{5}$

Step:3.2

Combine the fractions on the right side.

$x = \frac{-34 - 1}{5}$

Step:3.3

Subtract $1$ from $-34$.

$x = \frac{-35}{5}$

Step:3.4

Simplify the fraction to find $x$.

$x = -7$

Knowledge Notes:

1. Distributive Property: This property allows us to multiply a single term and two or more terms inside a set of parentheses. For example, $a(b + c) = ab + ac$.

2. Combining Like Terms: When simplifying expressions, we can combine terms that have the same variables raised to the same power. For example, $2x + 3x = 5x$.

3. Simplifying Fractions: To simplify a fraction, we can divide the numerator and the denominator by their greatest common factor. For example, $\frac{6}{8}$ simplifies to $\frac{3}{4}$ after dividing both the numerator and the denominator by $2$.

4. Multiplying Fractions: To multiply fractions, we multiply the numerators together and the denominators together. For example, $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

5. Negative Numbers: When dealing with negative numbers, remember that multiplying or dividing two negative numbers results in a positive number, while multiplying or dividing a positive number by a negative number results in a negative number.

6. Solving Linear Equations: The goal is to isolate the variable on one side of the equation. This often involves simplifying expressions, combining like terms, and performing inverse operations (such as adding to both sides to cancel out subtraction).