Twice the larger of two numbers is three more than five times the smaller and…

To solve a word problem describing two related unknown quantities, translate each verbal relationship into a linear equation in two variables, then solve the resulting system by substitution or elimination.

1. Let L denote the larger number and S denote the smaller number.

2. "Twice the larger is three more than five times the smaller" translates to 2L = 5S + 3.

3. "The sum of four times the larger and three times the smaller is 71" translates to 4L + 3S = 71.

4. Multiply the first equation by 2: 4L = 10S + 6.

5. Substitute this into the second equation: (10S + 6) + 3S = 71, which gives 13S + 6 = 71.

6. Solve for S: 13S = 65, so S = 5.

7. Substitute S = 5 into 2L = 5S + 3: 2L = 28, so L = 14.

Verify both conditions with L = 14 and S = 5: twice the larger, 2 x 14 = 28, equals five times the smaller plus three, 5 x 5 + 3 = 28. Also, four times the larger plus three times the smaller, 4 x 14 + 3 x 5 = 56 + 15 = 71, matches the given total. Both equations are satisfied, so the numbers are 14 and 5.