twice a number minus a second number is -1. twice the second number added to three times the first number is 9. find the two numbers.
first number --- x
2nd number --- y
2x - y = -1
or
y = 2x+1
2y + 3x = 9
2(2x+1) + 3x = 9
4x + 2 + 3x = 9
7x = 7
x = 1
then y = 3
The first number is 1, and the second number is 3
check:
twice the first - second
= 2 - 3 = -1 , check!
twice the second added to three times the first
2(3) + 3(1)
= 9 , check!
qed
Let's assume the first number is "x" and the second number is "y".
According to the given information:
1) "Twice a number minus a second number is -1":
2x - y = -1 (Equation 1)
2) "Twice the second number added to three times the first number is 9":
3x + 2y = 9 (Equation 2)
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of elimination.
Multiply Equation 1 by 2 to make the coefficients of "y" the same:
2(2x - y) = 2(-1)
4x - 2y = -2 (Equation 3)
Now, we can eliminate "y" by subtracting Equation 3 from Equation 2:
(3x + 2y) - (4x - 2y) = 9 - (-2)
3x + 2y - 4x + 2y = 9 + 2
-x + 4y = 11 (Equation 4)
Now we have two equations:
Equation 3: 4x - 2y = -2
Equation 4: -x + 4y = 11
Let's multiply Equation 4 by 2 to make the coefficients of "x" the same:
-2(-x + 4y) = 2(11)
2x - 8y = 22 (Equation 5)
Now, we can eliminate "x" by adding Equation 5 to Equation 3:
(4x - 2y) + (2x - 8y) = -2 + 22
6x - 10y = 20
Divide this equation by 2:
3x - 5y = 10 (Equation 6)
Now, we have two equations:
Equation 5: 6x - 10y = 20
Equation 6: 3x - 5y = 10
Let's multiply Equation 6 by 2 to make the coefficients of "x" the same:
2(3x - 5y) = 2(10)
6x - 10y = 20 (Equation 7)
Now, we can eliminate "x" by subtracting Equation 6 from Equation 7:
(6x - 10y) - (6x - 10y) = 20 - 20
0 = 0
Since the result is 0, it means that both equations represent the same line, which means they are dependent or coincident. This implies that the system of equations has infinitely many solutions.
Therefore, the two numbers cannot be determined uniquely.
To solve this problem, let's assign variables to the two numbers. Let's call the first number "x" and the second number "y."
From the given information, we can create two equations:
Equation 1: Twice a number minus a second number is -1.
2x - y = -1
Equation 2: Twice the second number added to three times the first number is 9.
3x + 2y = 9
Now, we can solve the system of equations using different methods such as substitution or elimination. Let's use the elimination method:
1. Multiply the first equation by 3 and the second equation by 2 to make the coefficient for "y" equal in both equations:
Equation 1: (2)(2x - y) = (2)(-1)
4x - 2y = -2
Equation 2: (3)(3x + 2y) = (3)(9)
9x + 6y = 27
2. Multiply the first equation by 3 and subtract it from the second equation to eliminate "y":
12x - 6y = -6
9x + 6y = 27
--------------
21x = 21
3. Divide both sides of the equation by 21 to solve for "x":
x = 1
4. Substitute the value of "x" back into one of the original equations to solve for "y." Let's use Equation 1:
2x - y = -1
2(1) - y = -1
2 - y = -1
-y = -1 - 2
-y = -3
y = 3
So, the two numbers are x = 1 and y = 3.