SOLUTION: i don't understand how to solve problems where it says consistent and inconsistent systems. for example it says: Solve each system algebraically. Identify each system as consist

Algebra ->  Systems-of-equations -> SOLUTION: i don't understand how to solve problems where it says consistent and inconsistent systems. for example it says: Solve each system algebraically. Identify each system as consist      Log On


   

Question 130819This question is from textbook Algebra 1
: i don't understand how to solve problems where it says consistent and inconsistent systems.
for example it says: Solve each system algebraically. Identify each system as consistent and dependent, consistent and independent, or inconsistent.
2f+3g=11
f-g=-7
i mostly don't undestand the differences between consistent dependent, consistent independent, or inconsistent, but this problem confuses me. This question is from textbook Algebra 1

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Start with the given system of equations:

system%282f%2B3g=11%2Cf-g=-7%29



Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for g.





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for g, we would have to eliminate f (or vice versa).


So lets eliminate f. In order to do that, we need to have both f coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the f coefficients equal in magnitude but opposite in sign, we need to multiply both f coefficients by some number to get them to an common number. So if we wanted to get 2 and 1 to some equal number, we could try to get them to the LCM.



Since the LCM of 2 and 1 is 2, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -2 like this:




1%282f%2B3g%29=1%2811%29 Multiply the top equation (both sides) by 1
-2%28f-g%29=-2%28-7%29 Multiply the bottom equation (both sides) by -2




Distribute and multiply

2f%2B3g=11
-2f%2B2g=14


Now add the equations together. In order to add 2 equations, group like terms and combine them

%282f-2f%29%2B%283g%2B2g%29=11%2B14

Combine like terms and simplify



cross%282f-2f%29%2B5g=25 Notice how the f terms cancel out




5g=25 Simplify




g=25%2F5 Divide both sides by 5 to isolate g




g=5 Reduce



Now plug this answer into the top equation 2f%2B3g=11 to solve for x

2f%2B3g=11 Start with the first equation



2f%2B3%285%29=11 Plug in g=5




2f%2B15=11 Multiply



2f=11-15Subtract 15 from both sides


2f=-4 Combine like terms on the right side


f=%28-4%29%2F%282%29 Divide both sides by 2 to isolate f



f=-2 Divide




So our answer is
f=-2 and g=5



which also looks like




So the system is independent and consistent