Find x if (19x + 83) divided by 7 equals 8 squared minus 5 squared.

Step-by-step solution

1. 1

   Evaluate the right-hand side first. $8^2-5^2 = 64-25 = 39$. The difference-of-squares identity gives the same thing faster: $(8-5)(8+5) = 3\cdot13 = 39$.

2. 2

   Rewrite the equation. $(19x+83):7 = 39.$ The unknown is inside the dividend, so the inverse operation is multiplication.

3. 3

   Undo the division by 7. $19x+83 = 39\cdot7 = 273.$

4. 4

   Isolate the x-term. $19x = 273-83 = 190.$

5. 5

   Divide by 19. $x = \frac{190}{19} = 10,$ since $19\cdot10 = 190$ exactly.

6. 6

   Check in the original equation. $19(10)+83 = 190+83 = 273$, and $273:7 = 39$, which equals $8^2-5^2$. The answer is confirmed.