Please solve this question---**Question Stem:** Solve the following equation for z: **Mathematical Formula:** z/3 - 5/7 = z/5 - 1/8 **Prompt for Answer:** Z = [blank box for answer]

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Let's solve this linear equation with fractions. We have z over 3 minus 5 over 7 equals z over 5 minus 1 over 8. This is a linear equation that requires careful algebraic manipulation. Our strategy is to move all z terms to the left side and all constants to the right side, then find common denominators and solve for z. Let's rearrange the terms systematically. Starting with our original equation, we need to move z over 5 from the right side to the left side, and move 5 over 7 from the left side to the right side. When we move terms across the equals sign, we must change their signs. This gives us z over 3 minus z over 5 equals negative 1 over 8 plus 5 over 7. Now we have all z terms on the left and all constants on the right. Now we need to find common denominators to combine the fractions. For the left side, we have z over 3 minus z over 5. The least common denominator of 3 and 5 is 15. So z over 3 becomes 5z over 15, and z over 5 becomes 3z over 15. This gives us 2z over 15. For the right side, we have negative 1 over 8 plus 5 over 7. The least common denominator of 8 and 7 is 56. So negative 1 over 8 becomes negative 7 over 56, and 5 over 7 becomes 40 over 56. This gives us 33 over 56. Our simplified equation is 2z over 15 equals 33 over 56. Now we solve for z by isolating it. Starting with 2z over 15 equals 33 over 56, we multiply both sides by 15 over 2 to cancel the coefficient of z. This gives us z equals 33 over 56 times 15 over 2. Multiplying the fractions, we get z equals 33 times 15 over 56 times 2, which equals 495 over 112. We can check if this fraction can be simplified by finding the greatest common divisor of 495 and 112, which is 7. However, 495 divided by 7 is not a whole number, so our final answer is z equals 495 over 112. Let's verify our solution by substituting z equals 495 over 112 back into the original equation. For the left side, we have 495 over 112 divided by 3 minus 5 over 7. This equals 495 over 336 minus 240 over 336, which gives us 255 over 336. For the right side, we have 495 over 112 divided by 5 minus 1 over 8. This equals 495 over 560 minus 70 over 560, which simplifies to 425 over 560 or 255 over 336. Since both sides equal 255 over 336, our solution is verified. Therefore, z equals 495 over 112, which is approximately 4.42.