The sum of two numbers is 103, the larger number is 1 more than 5 times the smaller number.
use x=smaller number
y=larger number
To solve this problem, we can use a system of equations:
Let x be the smaller number.
Let y be the larger number.
Based on the given information, we can write the following equations:
Equation 1: x + y = 103 (the sum of two numbers is 103)
Equation 2: y = 5x + 1 (the larger number is 1 more than 5 times the smaller number)
To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method.
Step 1: Solve Equation 2 for y in terms of x:
y = 5x + 1
Step 2: Substitute the value of y in Equation 1:
x + (5x + 1) = 103
Step 3: Simplify and solve for x:
6x + 1 = 103
6x = 102
x = 17
Step 4: Substitute the value of x back into Equation 2 to find the value of y:
y = 5(17) + 1
y = 86 + 1
y = 87
Therefore, the smaller number is 17 and the larger number is 87.
To solve this problem, we can set up a system of equations based on the given information.
Let's assign x to be the smaller number and y to be the larger number.
1. The sum of two numbers is 103:
x + y = 103 ---- (equation 1)
2. The larger number is 1 more than 5 times the smaller number:
y = 5x + 1 ---- (equation 2)
Now, we have a system of two equations with two variables. We can solve them simultaneously to find the values of x and y.
Method 1: Substitution Method
We can solve equation 1 for x and then substitute it into equation 2.
From equation 1, we can solve for x:
x = 103 - y
Substituting this value of x into equation 2:
y = 5(103 - y) + 1
y = 515 - 5y + 1
6y = 516
y = 86
Now, substitute y = 86 back into equation 1 to solve for x:
x + 86 = 103
x = 103 - 86
x = 17
Therefore, the smaller number (x) is 17 and the larger number (y) is 86.
Method 2: Elimination Method
We can eliminate one of the variables by adding or subtracting the two equations.
Multiply equation 1 by 5 to make the coefficients of y in both equations the same:
5(x + y) = 5(103)
5x + 5y = 515 ---- (equation 3)
Now subtract equation 2 from equation 3:
(5x + 5y) - (y) = 515 - (5x + 1)
5x + 5y - y = 515 - 5x - 1
4y = 514 - 5x
Since we already have x = 103 - y, we can substitute x in the above equation:
4y = 514 - 5(103 - y)
4y = 514 - 515 + 5y
4y - 5y = -1
-y = -1
y = 1
Now, substitute y = 1 back into equation 1 to solve for x:
x + 1 = 103
x = 103 - 1
x = 102
Thus, the smaller number (x) is 102 and the larger number (y) is 1.
Therefore, there are two possible solutions to this problem:
Case 1: x = 17, y = 86
Case 2: x = 102, y = 1
y = (1 + 5 x)
x + y = 103
x + (1 + 5 x) = 103
6 x + 1 = 103
6 x = 102
x = 17
y = (1 + 5 x) = 1 + 85 = 86
check
17 + 86 = ?