Final answer:
To solve the problem, we can set up a system of equations by writing equations for the sum and difference of the two numbers. By using the method of substitution, we can solve for the values of x and y, which represent the larger and smaller numbers. The solution to the system of equations is x = 72 and y = 10.
Step-by-step explanation:
To solve this problem, we can use a system of equations. Let x be the larger number and y be the smaller number. The problem states that the sum of the two numbers is 82, so we can write the equation x + y = 82. The problem also states that the difference between the two numbers is 62, so we can write the equation x - y = 62.
To solve the system of equations, we can use the method of substitution. We can solve one equation for one variable and substitute it into the other equation. Let's solve the first equation for x: x = 82 - y. Now we can substitute this expression for x in the second equation: (82 - y) - y = 62. Simplifying the equation, we get 82 - 2y = 62. Subtracting 82 from both sides, we have -2y = -20. Dividing both sides by -2, we get y = 10. Now that we know the value of y, we can substitute it back into one of the original equations to find x. Substituting y = 10 into the equation x + y = 82, we get x + 10 = 82. Subtracting 10 from both sides, we have x = 72. Therefore, the two numbers are 72 and 10.