Seven times the smaller of 2 numbers plus 9 times the larger is 178. When 10 times the larger is increased by 11 times the smaller the result is 230. Find the numbers.
Let's assume the smaller number as 'x' and the larger number as 'y'.
According to the first statement, "Seven times the smaller of 2 numbers plus 9 times the larger is 178", we can write the equation as:
7x + 9y = 178 ----(1)
According to the second statement, "When 10 times the larger is increased by 11 times the smaller, the result is 230", we can write the equation as:
10y + 11x = 230 ----(2)
To solve these equations, we will use the method of substitution.
First, solve equation (1) for x in terms of y:
7x = 178 - 9y
x = (178 - 9y) / 7 ----(3)
Now substitute the value of x from equation (3) into equation (2):
10y + 11((178 - 9y) / 7) = 230
Multiply both sides of the equation by 7 to eliminate the fraction:
70y + 11(178 - 9y) = 1610
Distribute 11:
70y + 1958 - 99y = 1610
Combine like terms:
-29y + 1958 = 1610
Subtract 1958 from both sides:
-29y = -348
Now, divide by -29 on both sides to solve for y:
y = (-348) / (-29)
y = 12
Substitute the value of y back into equation (3) to solve for x:
x = (178 - 9(12)) / 7
x = (178 - 108) / 7
x = 70 / 7
x = 10
Therefore, the smaller number is 10 and the larger number is 12.
To solve this problem, let's start by assigning variables to the unknowns. Let's call the smaller number "x" and the larger number "y".
From the given information and using the variables, we can translate the first sentence into an equation:
7x + 9y = 178 ---(Equation 1)
Now, let's move on to the second sentence:
"When 10 times the larger is increased by 11 times the smaller the result is 230."
This can be expressed as:
10y + 11x = 230 ---(Equation 2)
We now have a system of two equations with two variables (x and y). We can solve this system of equations using the method of substitution or elimination.
Let's solve it using the method of substitution:
1. Solve Equation 1 for x:
7x = 178 - 9y
x = (178 - 9y)/7
2. Substitute the value of x in Equation 2:
10y + 11((178 - 9y)/7) = 230
Now, we can solve this equation to find the value of y.
10y + (11/7)(178 - 9y) = 230
Multiply both sides by 7 to get rid of the fraction:
70y + 11(178 - 9y) = 1610
Expand the equation:
70y + 1958 - 99y = 1610
Combine like terms:
-29y + 1958 = 1610
Subtract 1958 from both sides:
-29y = -348
Divide both sides by -29:
y = 12
Now that we have found the value of y, we can substitute it back into Equation 1 to solve for x:
7x + 9(12) = 178
7x + 108 = 178
Subtract 108 from both sides:
7x = 70
Divide both sides by 7:
x = 10
Hence, the two numbers are x = 10 and y = 12.
If the numbers are x and y, then
7x + 9y = 178
10y + 11x = 230
Now just solve for x and y
x=3
y=5