Solve for v 5/14v-3/7=2/7

Solution:

Step 1:

Combine the term with $v$ and the constant on the left side of the equation.

$$\frac{5}{14}v - \frac{3}{7} = \frac{2}{7}$$

Step 2:

Isolate the variable term by moving constants to the opposite side.

Step 2.1:

Add $\frac{3}{7}$ to each side to eliminate the constant from the left.

$$\frac{5}{14}v = \frac{2}{7} + \frac{3}{7}$$

Step 2.2:

Combine like terms on the right side under a common denominator.

$$\frac{5}{14}v = \frac{2 + 3}{7}$$

Step 2.3:

Perform the addition in the numerator.

$$\frac{5}{14}v = \frac{5}{7}$$

Step 3:

Clear the fraction by multiplying each side by the reciprocal of the coefficient of $v$.

$$\frac{14}{5} \cdot \frac{5}{14}v = \frac{14}{5} \cdot \frac{5}{7}$$

Step 4:

Simplify both sides of the equation.

Step 4.1:

Simplify the left side.

Step 4.1.1:

Cancel out the common factors.

Step 4.1.1.1:

Eliminate the factor of $14$.

$$\frac{\cancel{14}}{5} \cdot \frac{5}{\cancel{14}}v = \frac{14}{5} \cdot \frac{5}{7}$$

Step 4.1.1.2:

Cancel the factor of $5$.

$$\frac{1}{\cancel{5}} \cdot \cancel{5}v = \frac{14}{5} \cdot \frac{5}{7}$$

Step 4.1.2:

Rewrite the simplified expression.

$$v = \frac{14}{5} \cdot \frac{5}{7}$$

Step 4.2:

Simplify the right side.

Step 4.2.1:

Cancel out the common factors.

Step 4.2.1.1:

Eliminate the factor of $7$.

$$v = \frac{2 \cdot \cancel{7}}{5} \cdot \frac{5}{\cancel{7}}$$

Step 4.2.1.2:

Cancel the factor of $5$.

$$v = \frac{2}{\cancel{5}} \cdot \cancel{5}$$

Step 4.2.2:

Rewrite the simplified expression.

$$v = 2$$

Knowledge Notes:

1. Combining Like Terms: When solving equations, we combine terms with the same variables or constants to simplify the equation.

2. Isolating the Variable: The goal in solving an equation is to isolate the variable on one side of the equation to find its value.

3. Clearing Fractions: Multiplying both sides of an equation by the reciprocal of a fraction eliminates the fraction, making it easier to solve.

4. Simplifying Expressions: This involves canceling out common factors on both sides of an equation to reduce it to its simplest form.

5. Reciprocal: The reciprocal of a number is 1 divided by that number. For a fraction, it is obtained by swapping the numerator and the denominator.

6. Equation Solving Process: The process typically involves simplifying the equation, isolating the variable, and then solving for the variable by performing arithmetic operations.