When two numbers are added together, the result is 45.Twice the greater number is six more than five times the lesser number. What are the numbers.
a+b=45
2a=5b+6, 2a-5b=6
Multiply the top equation by -2.
-2a-2b=-90
2a-5b=6
Add the equations together.
-7b=-84
b=12
a=33
Let's solve this step-by-step.
Let's assume one number as 'x' and the other number as 'y'.
1. "When two numbers are added together, the result is 45."
That means x + y = 45. (Equation 1)
2. "Twice the greater number is six more than five times the lesser number."
Translation: 2x = 5y + 6. (Equation 2)
Now, we have two equations with two variables. We can solve them using a method called substitution:
From Equation 1, x = 45 - y.
Plugging this value of x into Equation 2:
2(45 - y) = 5y + 6.
Now, let's solve for y:
90 - 2y = 5y + 6.
Rearranging the equation:
90 - 6 = 5y + 2y,
84 = 7y,
y = 12.
Now that we have the value of y, we can substitute it back into Equation 1 to find the value of x:
x + 12 = 45,
x = 45 - 12,
x = 33.
So, the two numbers are 33 and 12.
To solve this problem, you need to set up a system of equations and solve them simultaneously. Let's assign variables to the unknown numbers.
Let's say the greater number is represented by 'x,' and the lesser number is represented by 'y.'
According to the problem, when the two numbers are added together, the result is 45. So we can set up the equation:
x + y = 45 ---- Equation 1
It is also given that twice the greater number (2x) is six more than five times the lesser number (5y). So we can set up the second equation:
2x = 5y + 6 ---- Equation 2
Now we have a system of equations, Equation 1 and Equation 2, that we can solve simultaneously to find the values of x and y.
One way to solve this system is by substitution. Let's solve Equation 1 for x:
x = 45 - y
Now let's substitute this value for x in Equation 2:
2(45 - y) = 5y + 6
90 - 2y = 5y + 6
90 - 6 = 5y + 2y
84 = 7y
y = 12
Now that we have the value of y, we can substitute it back into Equation 1 to find the value of x:
x + 12 = 45
x = 45 - 12
x = 33
Therefore, the greater number is 33, and the lesser number is 12.