the difference of twice the smaller integer and 7 times a larger is 4. when 5 times the larger integer is subtracted from 3 times the smaller, the result is -5 find marks the number
a = larger number
b = smaller number
The difference of twice the smaller integer and 7 times a larger is 4 means:
2 b - 7 a = 4
When 5 times the larger integer is subtracted from 3 times the smaller, the result is - 5 means:
3 b - 5 a = - 5
Now you must solve system of two equation:
2 b - 7 a = 4
3 b - 5 a = - 5
The solutios are:
a = - 2 , b = - 5
larger number = - 2
smaller number = - 5
Let's assume the smaller integer is x and the larger integer is y.
According to the first statement:
Twice the smaller integer minus 7 times the larger integer is equal to 4.
So we have the equation:
2x - 7y = 4
According to the second statement:
3 times the smaller integer minus 5 times the larger integer is equal to -5.
So we have the equation:
3x - 5y = -5
Now we have a system of two equations with two variables:
2x - 7y = 4 ...(1)
3x - 5y = -5 ...(2)
We can solve this system of equations using the method of substitution or elimination.
Let's use the method of substitution:
From equation (1), we can express x in terms of y:
2x = 7y + 4
x = (7y + 4)/2
Substitute this value of x into equation (2):
3[(7y + 4)/2] - 5y = -5
(21y + 12 - 10y)/2 - 5y = -5
11y + 12 - 10y - 10y = -5
y + 12 = -5
Subtract 12 from both sides:
y = -5 - 12
y = -17
Now we can substitute this value of y back into equation (1) to find x:
2x - 7(-17) = 4
2x + 119 = 4
2x = 4 - 119
2x = -115
x = -115/2
x = -57.5
Therefore, the smaller integer is -57.5 and the larger integer is -17.
To solve this problem, let's start by assigning variables to the smaller and larger integers. Let's call the smaller integer "x" and the larger integer "y".
According to the first statement, "the difference of twice the smaller integer and 7 times the larger is 4." Mathematically, we can express this as:
2x - 7y = 4 (Equation 1)
According to the second statement, "when 5 times the larger integer is subtracted from 3 times the smaller, the result is -5." Mathematically, we can express this as:
3x - 5y = -5 (Equation 2)
Now, we have a system of two equations with two unknowns (x and y). We can solve this system to find the values of x and y.
To do this, we can use a method called "substitution method." Let's solve Equation 2 for x:
3x - 5y = -5
3x = 5y - 5
x = (5y - 5) / 3 (Equation 3)
Now, substitute the value of x from Equation 3 into Equation 1:
2x - 7y = 4
2((5y - 5) / 3) - 7y = 4
(10y - 10) / 3 - 7y = 4
(10y - 10) - 21y = 4 * 3
10y - 10 - 21y = 12
-11y - 10 = 12
-11y = 22
y = -2
Now, substitute the value of y = -2 back into Equation 3 to find x:
x = (5y - 5) / 3
x = (5(-2) - 5) / 3
x = (-10 - 5) / 3
x = -15 / 3
x = -5
Therefore, the smaller integer (x) is -5, and the larger integer (y) is -2.