The sum of two numbers is 98. Their difference is 22. Write a system of equations that describes this situation. Solve by elimination to find the two numbers.
A.
x+y=98
x-y=22
55 and 33
B. x+y=98
x-y=22
60 and 38
C.
x-y=98
x+y=22
59 and 39
D. x+y=22
y-x=98
55 and 39
Answer (2)
Define the two numbers as x and y , and write the system of equations: x + y = 98 and x − y = 22 .
Add the two equations to eliminate y : 2 x = 120 .
Solve for x : x = 60 .
Substitute x = 60 into one of the original equations and solve for y : y = 38 . The two numbers are 60 and 38 .
Explanation
Problem Analysis Let's analyze the problem. We are given two pieces of information about two numbers: their sum and their difference. We need to find the numbers themselves. We will use a system of equations to represent the given information and solve it using the elimination method.
Setting up the Equations Let x and y be the two numbers. We can write the following system of equations:
x + y = 98 x − y = 22
Elimination Method To solve this system by elimination, we can add the two equations together. Notice that the y terms will cancel out:
( x + y ) + ( x − y ) = 98 + 22 2 x = 120
Solving for x Now, we can solve for x by dividing both sides by 2:
x = 2 120 x = 60
Solving for y Next, we substitute the value of x back into one of the original equations to solve for y . Let's use the first equation:
60 + y = 98
Finding y Subtract 60 from both sides to isolate y :
y = 98 − 60 y = 38
The Solution So, the two numbers are 60 and 38.
Examples
Imagine you're managing a small business and need to track income and expenses. If your total revenue plus your total expenses equals $98, and the difference between your revenue and expenses is $22, you can use a system of equations to determine your exact revenue and expenses. This method is useful for balancing budgets, managing inventory, and making financial forecasts.
The two numbers that sum to 98 and have a difference of 22 are 60 and 38. We set up the equations x + y = 98 and x − y = 22 , solved using elimination, and found the values. Therefore, the correct answer is option B.
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