Solve $\frac{z-1}{10}=\frac{z+1}{12}$.
Answer (1)
Multiply both sides of the equation by 60 to eliminate the fractions.
Distribute the constants: 6 z − 6 = 5 z + 5 .
Isolate z by subtracting 5 z from both sides and adding 6 to both sides.
The solution is 11 .
Explanation
Problem Analysis We are given the equation 10 z − 1 = 12 z + 1 and asked to solve for z .
Eliminating Fractions To solve the equation, we first multiply both sides by the least common multiple (LCM) of 10 and 12. The LCM of 10 and 12 is 60. Multiplying both sides by 60, we get: 60 ⋅ 10 z − 1 = 60 ⋅ 12 z + 1 Simplifying, we have: 6 ( z − 1 ) = 5 ( z + 1 )
Expanding the Equation Next, we distribute the constants on both sides of the equation: 6 z − 6 = 5 z + 5
Isolating z Now, we want to isolate z . Subtract 5 z from both sides: 6 z − 5 z − 6 = 5 z − 5 z + 5 z − 6 = 5
Solving for z Finally, add 6 to both sides to solve for z :
z − 6 + 6 = 5 + 6 z = 11
Final Answer Therefore, the solution to the equation is z = 11 .
Examples
Imagine you're adjusting the settings on two different machines to produce the same output. This problem is similar to finding the right input value (z) that makes the output rates of both machines equal, even though they have different scaling factors (10 and 12). Understanding how to solve such equations is crucial in engineering and manufacturing to ensure consistent performance across different systems.
