The sum of two numbers is 19, and their difference is 55. What are the two numbers?
Answer (2)
Define the two numbers as x and y , and set up the equations x + y = 19 and x − y = 55 .
Add the two equations to eliminate y and solve for x : 2 x = 74 , so x = 37 .
Substitute the value of x into one of the original equations to solve for y : 37 + y = 19 , so y = − 18 .
The two numbers are 37 , − 18 .
Explanation
Setting up the equations Let's call the two numbers x and y . We are given two pieces of information:
The sum of the two numbers is 19, which we can write as an equation: x + y = 19 .
The difference of the two numbers is 55, which we can write as an equation: x − y = 55 .
We now have a system of two equations with two variables. We can solve this system to find the values of x and y .
Eliminating y We can use the method of elimination to solve this system. Add the two equations together:
( x + y ) + ( x − y ) = 19 + 55
2 x = 74
Solving for x Now, divide both sides by 2 to solve for x :
x = 2 74 = 37
Substituting x to find y Now that we have the value of x , we can substitute it into either of the original equations to solve for y . Let's use the first equation:
37 + y = 19
Solving for y Subtract 37 from both sides to solve for y :
y = 19 − 37 = − 18
Stating the answer So, the two numbers are x = 37 and y = − 18 .
Examples
Imagine you're managing a budget. You know your total income and how much more you spend than you earn. This problem is like figuring out exactly how much you earn and how much you spend. By setting up a system of equations, you can determine these two unknown amounts, helping you to better manage your finances. This method is useful in various scenarios, from balancing accounts to optimizing resource allocation.
The two numbers are 37 and -18. This is determined by solving the equations set up from the sum and difference provided. The equations were manipulated to find the values of x and y successfully.
;
