Questions: Solve the following equation. Express your answer as an integer, simplified fraction, or decimal rounded to two decimal places. 7/6 z + 7/10 z - 13/30 = 15/2 z=

Solution Steps

Combine the terms involving \( z \) on the left side of the equation: \[ \frac{7}{6} z + \frac{7}{10} z = \left( \frac{7}{6} + \frac{7}{10} \right) z \] To add the fractions, find a common denominator. The least common denominator (LCD) of 6 and 10 is 30: \[ \frac{7}{6} = \frac{35}{30}, \quad \frac{7}{10} = \frac{21}{30} \] Now add them: \[ \frac{35}{30} z + \frac{21}{30} z = \frac{56}{30} z \]

Add \( \frac{13}{30} \) to both sides of the equation to isolate the terms with \( z \): \[ \frac{56}{30} z = \frac{15}{2} + \frac{13}{30} \] Convert \( \frac{15}{2} \) to a fraction with a denominator of 30: \[ \frac{15}{2} = \frac{225}{30} \] Now add the fractions: \[ \frac{56}{30} z = \frac{225}{30} + \frac{13}{30} = \frac{238}{30} \]

Divide both sides of the equation by \( \frac{56}{30} \) to solve for \( z \): \[ z = \frac{\frac{238}{30}}{\frac{56}{30}} = \frac{238}{30} \times \frac{30}{56} = \frac{238}{56} \] Simplify the fraction: \[ z = \frac{238 \div 14}{56 \div 14} = \frac{17}{4} \]

Final Answer