Five times the larger of two numbers plus four times the smaller is 271. Three times the larger less twice the smaller is 57. find the numbers
5L+4S=271
3L-2S=57
double the lower equation, then add the two equations.
11L=271+114
L=35
S=3*35/2-57/2=(105-57)/2=24
Let's represent the larger number as "x" and the smaller number as "y".
According to the given information, we have two equations:
Equation 1: 5x + 4y = 271
Equation 2: 3x - 2y = 57
To solve this system of equations, we can use either the substitution method or the elimination method. Let's use the elimination method:
Multiply Equation 1 by 2 and Equation 2 by 3 to eliminate the "y" variable:
Equation 3: 10x + 8y = 542
Equation 4: 9x - 6y = 171
Add Equation 3 and Equation 4 together:
(10x + 8y) + (9x - 6y) = 542 + 171
19x + 2y = 713
Now, we have two equations:
Equation 5 (from adding Equation 3 and Equation 4): 19x + 2y = 713
Equation 2: 3x - 2y = 57
Add Equation 5 and Equation 2 together:
(19x + 2y) + (3x - 2y) = 713 + 57
22x = 770
x = 770 / 22
x ≈ 35
Substitute x = 35 into Equation 2 to find y:
3(35) - 2y = 57
105 - 2y = 57
-2y = 57 - 105
-2y = -48
y = -48 / -2
y = 24
Therefore, the larger number (x) is approximately 35 and the smaller number (y) is 24.
To solve this problem, let's define our unknown variables. Let's call the larger number "L" and the smaller number "S."
According to the given information, we have two equations:
1) Five times the larger of two numbers plus four times the smaller is 271:
5L + 4S = 271
2) Three times the larger number less twice the smaller is 57:
3L - 2S = 57
To solve these equations, we can use a method called "substitution."
First, let's solve the second equation for L:
3L - 2S = 57
3L = 57 + 2S
L = (57 + 2S) / 3
Now, substitute this value of L into the first equation:
5L + 4S = 271
5((57 + 2S) / 3) + 4S = 271
Now, we can simplify and solve for S:
5(57 + 2S) / 3 + 4S = 271
(285 + 10S) / 3 + 4S = 271
285 + 10S + 12S = 813
22S = 813 - 285
22S = 528
S = 528 / 22
S = 24
Now that we have the value of S, we can substitute it back into the second equation to find L:
3L - 2S = 57
3L - 2(24) = 57
3L - 48 = 57
3L = 57 + 48
3L = 105
L = 105 / 3
L = 35
So, the larger number (L) is 35, and the smaller number (S) is 24.