The sum of two numbers is 30 and their difference is 2. Find the two numbers by writing and solving a system of equations.
Equations: Substitution
x + y = 30
x = y - 2
To find the two numbers, we can set up a system of equations using the given information.
Let's denote the first number as 'x' and the second number as 'y'.
From the information given, we can write two equations:
Equation 1: The sum of two numbers is 30.
x + y = 30
Equation 2: Their difference is 2.
x - y = 2
Now, we have a system of equations. To solve this system, we can use the method of substitution or elimination.
Method 1: Substitution
Solve Equation 2 for x in terms of y:
x = 2 + y
Substitute this expression for x into Equation 1:
(2 + y) + y = 30
2y + 2 = 30
2y = 30 - 2
2y = 28
y = 28/2
y = 14
Now, substitute the value of y back into Equation 2 to solve for x:
x - 14 = 2
x = 2 + 14
x = 16
Therefore, the two numbers are 16 and 14.
Method 2: Elimination
Multiply Equation 2 by -1 to reverse the sign:
-x + y = -2
Add this equation to Equation 1:
(x + y) + (-x + y) = 30 + (-2)
2y = 28
y = 28/2
y = 14
Substitute the value of y back into Equation 1 to solve for x:
x + 14 = 30
x = 30 - 14
x = 16
Thus, we get the same result, which is 16 and 14 as the two numbers.