the sum of two numbers is 90.three times the larger decreased by twice the smaller is 160.what are the numbers?
one number is x, the other 90-x. Let x be the larger.
3x - 2(90-x) = 160
3x - 180 + 2x = 160
5x = 340
x = 68
so, the two numbers are 68 and 22
Check:
3*68 - 2*22 = 204-44 = 160
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To solve this problem, let's assign variables to the two numbers mentioned. Let's call the larger number "x" and the smaller number "y."
From the problem, we know two things:
1) The sum of the two numbers is 90, which means x + y = 90.
2) Three times the larger number decreased by twice the smaller number is equal to 160, which can be expressed as 3x - 2y = 160.
Now we have a system of two equations:
Equation 1: x + y = 90
Equation 2: 3x - 2y = 160
We can solve this system of equations using various methods, such as substitution or elimination.
Let's solve it using the elimination method:
1) Multiply Equation 1 by 3 to make the coefficients of "y" in both equations the same:
3(x + y) = 3(90)
3x + 3y = 270
2) Rearrange Equation 2 and multiply it by 3:
3x - 2y = 160
3(3x - 2y) = 3(160)
9x - 6y = 480
3) Now we can subtract the two equations:
(3x + 3y) - (9x - 6y) = 270 - 480
3x +3y - 9x + 6y = -210
-6x + 9y = -210
4) Divide the entire equation by -3:
(-6x + 9y) / -3 = -210 / -3
2x - 3y = 70
Now we have a new equation:
Equation 3: 2x - 3y = 70
5) Solve Equation 3 simultaneously with Equation 1:
x + y = 90
2x - 3y = 70
Using elimination or substitution, we can solve this new system of equations to find the values of x and y.