Please show me how to set and solve:
The sum of two numbers is 90. Their difference is 12. What are the numbers?
x + (x-12) = 90
2x = 102
x = 51
the two numbers are 39 and 51
To solve this problem, we will set up a system of equations based on the given information and then solve it. Let's call the two numbers x and y.
From the first statement, "The sum of two numbers is 90," we can set up the equation:
x + y = 90
From the second statement, "Their difference is 12," we can set up the equation:
x - y = 12
We now have a system of two equations:
Equation 1: x + y = 90
Equation 2: x - y = 12
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.
From Equation 1, we can solve for x:
x = 90 - y
Now, substitute this expression for x in Equation 2:
(90 - y) - y = 12
Simplify the equation:
90 - 2y = 12
Next, isolate the y variable:
-2y = 12 - 90
-2y = -78
Divide both sides by -2 to solve for y:
y = -78 / -2
y = 39
Now that we have the value of y, substitute it back into Equation 1 to find x:
x + 39 = 90
Subtract 39 from both sides:
x = 90 - 39
x = 51
So, the two numbers are x = 51 and y = 39.
To check the solution, you can substitute these values into both equations and see if they hold true.