maximize p=6x+10y
minimize p=14x+9y
use elimination to solve:
4x+5y=41
7x+5y=53
-4x-y=-16
-4x-5y=-32
2x-4=8
x+2y=9
9x-5x=13
4x-6y=2
To maximize or minimize a linear objective function, such as p, you need to find the values of x and y that satisfy the given constraints while optimizing the objective function.
1. To solve the system of equations using elimination:
- Start by subtracting the first equation from the second equation to eliminate the "y" term:
(7x + 5y) - (4x + 5y) = 53 - 41
Simplifying, you get:
7x + 5y - 4x - 5y = 12
Combine like terms:
3x = 12
Divide both sides by 3:
x = 4
- Substitute the value of x into either of the original equations to find y:
4x + 5y = 41
4(4) + 5y = 41
Simplifying, you get:
16 + 5y = 41
Subtract 16 from both sides:
5y = 25
Divide both sides by 5:
y = 5
Therefore, the solution for the system of equations 4x + 5y = 41 and 7x + 5y = 53 is x = 4 and y = 5.
2. To solve the system of equations -4x - y = -16 and -4x - 5y = -32 using elimination:
- Subtract the second equation from the first equation to eliminate the "y" term:
(-4x - y) - (-4x - 5y) = -16 - (-32)
Simplifying, you get:
-4x - y + 4x + 5y = 16 + 32
Combine like terms:
4y = 48
Divide both sides by 4:
y = 12
- Substitute the value of y into either of the original equations to find x:
-4x - y = -16
-4x - 12 = -16
Add 12 to both sides:
-4x = -4
Divide both sides by -4:
x = 1
Therefore, the solution for the system of equations -4x - y = -16 and -4x - 5y = -32 is x = 1 and y = 12.
3. To solve the system of equations 2x - 4 = 8 and x + 2y = 9:
- Solve the first equation for x:
2x - 4 = 8
Add 4 to both sides:
2x = 12
Divide both sides by 2:
x = 6
- Substitute the value of x into the second equation to find y:
6 + 2y = 9
Subtract 6 from both sides:
2y = 3
Divide both sides by 2:
y = 1.5
Therefore, the solution for the system of equations 2x - 4 = 8 and x + 2y = 9 is x = 6 and y = 1.5.
4. To solve the system of equations 9x - 5x = 13 and 4x - 6y = 2:
- Combine like terms in the first equation:
4x = 13
Divide both sides by 4:
x = 13/4
Simplify if needed.
- Substitute the value of x into the second equation to find y:
4(13/4) - 6y = 2
Simplify if needed.
13 - 6y = 2
Subtract 13 from both sides:
-6y = -11
Divide both sides by -6:
y = -11/-6
Simplify if needed.
Therefore, the solution for the system of equations 9x - 5x = 13 and 4x - 6y = 2 is x = 13/4 and y = -11/-6.