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.
[tex]
\begin{array}{l}
x+y=98 \\
x-y=22 \\
55 \text { and } 33
\end{array}
[/tex]
c.
[tex]
\begin{array}{l}
x-y=98 \\
x+y=22 \\
59 \text { and } 39
\end{array}
[/tex]
b.
[tex]
\begin{array}{l}
x+y=98 \\
x-y=22 \\
60 \text { and } 38
\end{array}
[/tex]
d.
[tex]
\begin{array}{l}
x+y=22 \\
y-x=98 \\
55 \text { and } 39
\end{array}
[/tex]
Please select the best answer from the choices provided
A
B
C
D
Answer (2)
Define the two numbers as x and y , and express the given information as a system of equations: x + y = 98 and x − y = 22 .
Solve the system of equations by adding the two equations to eliminate y : 2 x = 120 .
Find the value of x by dividing both sides by 2: x = 60 .
Substitute the value of x into one of the original equations to find y : y = 38 . The final answer is 60 and 38 .
Explanation
Problem Analysis Let's analyze the problem. We are given that the sum of two numbers is 98 and their difference is 22. We need to find these two numbers by setting up a system of equations and solving 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. This will eliminate the variable y :
( x + y ) + ( x − y ) = 98 + 22 2 x = 120
Solving for x Now, we can solve for x by dividing both sides of the equation by 2:
x = 2 120 = 60
Substituting x to find y Next, we substitute the value of x into one of the original equations to solve for y . Let's use the first equation:
60 + y = 98
Solving for y Subtract 60 from both sides to isolate y :
y = 98 − 60 = 38
Checking the Options So, the two numbers are 60 and 38. Now, let's check which of the given options matches our system of equations and the solution.
Option b has the correct system of equations and the correct solution: x + y = 98 , x − y = 22 , and the numbers are 60 and 38.
Final Answer Therefore, the correct answer is option b.
Examples
Imagine you're managing a small business and need to track your income and expenses. If your total revenue for the month is $98,000 and your profit (revenue minus expenses) is $22,000, you can use a system of equations to determine your exact expenses. By setting up the equations revenue + expenses = total and revenue - expenses = profit, you can solve for both revenue and expenses, helping you understand your business's financial health. This method is useful in various financial analyses, such as budgeting, cost estimation, and investment planning.
The system of equations for the two numbers is x + y = 98 and x − y = 22 . By using elimination, we find the two numbers are 60 and 38 . Therefore, the correct answer is option B.
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