Question

Which of the following ordered pairs represents the solution to the system given below?

2x + y = 20
4x − 2y = 40

A) (10,0)
B) (0,10)
C) (4,2)
D) (10, -10)

Answer 1

If you are under pressure try brute force:
Put for x the first number out of each of the pairs and the second for y into the first equation and see whether it fits. A successful candidate (for x and y) for the first equation is then used for the second as well. Once it has worked out for both you know which one it is.

Answer 2

For multiple choice it is often a quick solution. Be warned that your prof would probably prefer the elimination approach (Gauss) however.

Answer 3

And the bad retriceway of solving it is via formulating it as LP (linear programming) model. Here a sample (for Gurobi):

def main():
pass

if __name__ == '__main__':
main()

from gurobipy import *

m = Model("mip1")

x = m.addVar(vtype=GRB.CONTINUOUS, name="x", lb=-GRB.INFINITY, ub=GRB.INFINITY)
y = m.addVar(vtype=GRB.CONTINUOUS, name="y", lb=-GRB.INFINITY, ub=GRB.INFINITY)

m.update()

c1 = m.addConstr(2 * x + y == 20, name="1")
c2 = m.addConstr(4 * x - 2 * y == 40, name="2")

m.update()

m.setObjective(x, GRB.MAXIMIZE)

m.optimize()

print "_________________________________________________________"

if m.Status == GRB.OPTIMAL:
m.printQuality()
m.printAttr('x')
m.printAttr(['lb','ub'])

print('Obj: %g' % m.objVal)

print ('FeasibilityTol: %g' % m.Params.FeasibilityTol)

print "\nConstraint checkup:"

for c in m.getConstrs():
row = m.getRow(c)
LHS = row.getConstant()
for i in range(row.size()):
LHS += row.getCoeff(i) * row.getVar(i).X
if((c.Sense == '=')) & (abs(c.RHS-LHS)>m.Params.FeasibilityTol) | (c.Sense=='<') & (LHS>c.RHS+m.Params.FeasibilityTol) | ((c.Sense=='>') & (LHS<c.RHS-m.Params.FeasibilityTol)):
print "Violation in constraint " + c.ConstrName + ": LHS = " + str(LHS) + ", RHS = " + str(c.RHS)
else:
print "LHS: " + str(LHS)+ " consists of: "
for i in range(row.size()):
print " " + str(row.getCoeff(i)) + ' * ' + str(row.getVar(i).X)
print "________"
else:
print m.Status

Answer 4

Here is the output of your problem. I've replaced the value for X through XXXXXXX. Which way would you like to go?

*** Python 2.7.9 (default, Dec 10 2014, 12:28:03) [MSC v.1500 64 bit (AMD64)] on win32. ***
>>>
*** Remote Interpreter Reinitialized ***
>>>
[Dbg]>>>
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Coefficient statistics:
Matrix range [1e+00, 4e+00]
Objective range [1e+00, 1e+00]
Bounds range [0e+00, 0e+00]
RHS range [2e+01, 4e+01]
Presolve removed 2 rows and 2 columns
Presolve time: 0.05s
Presolve: All rows and columns removed
Iteration Objective Primal Inf. Dual Inf. Time
0 1.0000000e+01 0.000000e+00 0.000000e+00 0s

Solved in 0 iterations and 0.05 seconds
Optimal objective 1.000000000e+01
_________________________________________________________

Solution quality statistics for model mip1 :
Maximum violation (unscaled/scaled):
Bound : 0.00000000e+00 / 0.00000000e+00
Constraint : 0.00000000e+00 / 0.00000000e+00
Dual : 0.00000000e+00 / 0.00000000e+00

Variable x
-------------------------
x XXXXXXXXXXXXXX

Variable lb ub
--------------------------------------
x -1e+100 1e+100
y -1e+100 1e+100
Obj: 10
FeasibilityTol: 1e-06

Constraint checkup:
LHS: 20.0 consists of:
2.0 * 10.0
1.0 * 0.0
________
LHS: 40.0 consists of:
4.0 * 10.0
-2.0 * 0.0

Answer 5

im so lost

Answer 6

As a hint: By looking at the first equation it is obvious that you have to have for x something that when doubled would equals to 20 and at the same time the y should have no impact at all. The A is already quite good candidate therefore (y = 0 and x would give you 20 when doubled). Can you verify, you know just in case...

Answer 7

The test to verify that A is the solution could look like this:

A: x = 10; y = 0;

Your equations as follows:

2x + y = 20
4x − 2y = 40

We substitute the x and y through the values we suspect would fit:

2 * 10 + 0 = 20
4 * 10 − 2 * 0 = 40

Can you calculate and check that the terms are really equal to 20 and 40 resp? If do I would be confident that A is what you are looking for.

Answer 8

Have to go... I hope it helps. I will have a look in ~10 hours again. Good luck!

Answer 9

You just made it more confusing for the asker .-.

Answer 10

-2(2x + y = 20)
4x − 2y = 40
Keep in mind opposites cancel out :)
-4x-2y=-40
-4x-2y=-40

Answer 11

im really lost @kidrah69

Answer 12

We did the Elimination method

Answer 13

Opposites cancel :) do u kinda get it?

Answer 14

not really

Answer 15

Look at this
http://www.coolmath.com/algebra/12-2x2-systems-of-equations/03-solving-by-elimination-addition-01

Answer 16

get it? :-)