the difference of two numbers is 10. the sum of three times the smaller number and four times the larger is 75. find the numbers
so lets represent the numbers as variables
x will be the bigger number and y smaller.
x-y=10 3y+4x=75
solve for x
x-y=10 x=10+y
plug x into the other equation
3y+4(10+y)=75 3y+40+4y=75
solve for y and then plug it back into the equation to solve for x.
To solve this problem, let's assign variables to the unknown numbers. Let's call the smaller number "x" and the larger number "y".
From the information given, we have two equations:
1) The difference of two numbers is 10:
y - x = 10
2) The sum of three times the smaller number and four times the larger is 75:
3x + 4y = 75
We now have a system of two equations with two unknowns. We can solve this system of equations using substitution or elimination.
Let's solve it using the elimination method:
Multiply equation 1 by 4, so that both the x and y coefficients become 4:
4(y - x) = 4(10)
4y - 4x = 40
Next, we can subtract equation 2 from the new equation 3x + 4y = 75:
(3x + 4y) - (4y - 4x) = 75 - 40
3x + 4y - 4y + 4x = 35
7x = 35
x = 35 / 7
x = 5
Now, substitute the value of x back into equation 1 to find y:
y - 5 = 10
y = 10 + 5
y = 15
So, the solution to the problem is x = 5 and y = 15.