SOLUTION: Solve the system by using the substitution method. Determine whether the equations are Independent, Dependent, or Inconsistent: 1/2x + 1/3y = -1/3 1/2x + 2y = -7

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the system by using the substitution method. Determine whether the equations are Independent, Dependent, or Inconsistent: 1/2x + 1/3y = -1/3 1/2x + 2y = -7      Log On


   

Question 102273: Solve the system by using the substitution method. Determine whether the equations are Independent, Dependent, or Inconsistent:
1/2x + 1/3y = -1/3
1/2x + 2y = -7
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


%281%2F2%29%2Ax%2B%281%2F3%29%2Ay=-1%2F3 Start with the first equation


6%28%281%2F2%29%2Ax%2B%281%2F3%29%2Ay%29=%286%29%2A%28-1%2F3%29 Multiply both sides by the LCD 6



3%2Ax%2B2%2Ay=-2Distribute and simplify


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%281%2F2%29%2Ax%2B%282%29%2Ay=-7 Start with the second equation


2%28%281%2F2%29%2Ax%2B%282%29%2Ay%29=%282%29%2A%28-7%29 Multiply both sides by the LCD 2



1%2Ax%2B4%2Ay=-14 Distribute and simplify


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Lets start with the given system of linear equations

3%2Ax%2B2%2Ay=-2
1%2Ax%2B4%2Ay=-14

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

2%2Ay=-2-3%2AxSubtract 3%2Ax from both sides

y=%28-2-3%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=-1-%283%2F2%29%2Ax Now we've fully isolated y

Since y equals -1-%283%2F2%29%2Ax we can substitute the expression -1-%283%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B4%2Ahighlight%28%28-1-%283%2F2%29%2Ax%29%29=-14 Replace y with -1-%283%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B4%2A%28-1%29%2B4%28-3%2F2%29x=-14 Distribute 4 to -1-%283%2F2%29%2Ax

1%2Ax-4-%2812%2F2%29%2Ax=-14 Multiply



1%2Ax-4-6%2Ax=-14 Reduce any fractions

1%2Ax-6%2Ax=-14%2B4Add 4 to both sides


1%2Ax-6%2Ax=-10 Combine the terms on the right side



-5%2Ax=-10 Now combine the terms on the left side.


cross%28%281%2F-5%29%28-5%2F1%29%29x=%28-10%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5%2F1 and isolate x

So when we multiply -10%2F1 and 1%2F-5 (and simplify) we get



x=2 <---------------------------------One answer

Now that we know that x=2, lets substitute that in for x to solve for y

1%282%29%2B4%2Ay=-14 Plug in x=2 into the 2nd equation

2%2B4%2Ay=-14 Multiply

4%2Ay=-14-2Subtract 2 from both sides

4%2Ay=-16 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%28-16%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=-16%2F4 Multiply the terms on the right side


y=-4 Reduce


So this is the other answer


y=-4<---------------------------------Other answer


So our solution is

x=2 and y=-4

which can also look like

(2,-4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax%2B2%2Ay=-2
1%2Ax%2B4%2Ay=-14

we get


graph of 3%2Ax%2B2%2Ay=-2 (red) and 1%2Ax%2B4%2Ay=-14 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (2,-4). This verifies our answer.


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Check:

Plug in (2,-4) into the system of equations


Let x=2 and y=-4. Now plug those values into the equation 3%2Ax%2B2%2Ay=-2

3%2A%282%29%2B2%2A%28-4%29=-2 Plug in x=2 and y=-4


6-8=-2 Multiply


-2=-2 Add


-2=-2 Reduce. Since this equation is true the solution works.


So the solution (2,-4) satisfies 3%2Ax%2B2%2Ay=-2



Let x=2 and y=-4. Now plug those values into the equation 1%2Ax%2B4%2Ay=-14

1%2A%282%29%2B4%2A%28-4%29=-14 Plug in x=2 and y=-4


2-16=-14 Multiply


-14=-14 Add


-14=-14 Reduce. Since this equation is true the solution works.


So the solution (2,-4) satisfies 1%2Ax%2B4%2Ay=-14


Since the solution (2,-4) satisfies the system of equations


3%2Ax%2B2%2Ay=-2
1%2Ax%2B4%2Ay=-14


this verifies our answer.