SOLUTION: find the solution to the following system of linear equations by a) graphing b) substitution c) addition / subtraction method 4x - 3y = 2 -2x + 3y = -4

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: find the solution to the following system of linear equations by a) graphing b) substitution c) addition / subtraction method 4x - 3y = 2 -2x + 3y = -4      Log On


   

Question 166089: find the solution to the following system of linear equations by
a) graphing
b) substitution
c) addition / subtraction method
4x - 3y = 2
-2x + 3y = -4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)



Start with the given system of equations:


system%284x-3y=2%2C-2x%2B3y=-4%29


In order to graph these equations, we must solve for y first.


Let's graph the first equation:


4x-3y=2 Start with the first equation.


-3y=2-4x Subtract 4x from both sides.


y=%282-4x%29%2F%28-3%29 Divide both sides by -3 to isolate y.


y=%284%2F3%29x-2%2F3 Rearrange the terms and simplify.


Now let's graph the equation:


Graph of y=%284%2F3%29x-2%2F3.


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Now let's graph the second equation:


-2x%2B3y=-4 Start with the second equation.


3y=-4%2B2x Add 2x to both sides.


y=%28-4%2B2x%29%2F%283%29 Divide both sides by 3 to isolate y.


y=%282%2F3%29x-4%2F3 Rearrange the terms and simplify.


Now let's graph the equation:


Graph of y=%282%2F3%29x-4%2F3.


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Now let's graph the two equations together:


Graph of y=%284%2F3%29x-2%2F3 (red). Graph of y=%282%2F3%29x-4%2F3 (green)


From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is . This tells us that the system of equations is consistent and independent.





b)





Start with the given system of equations:

system%284x-3y=2%2C-2x%2B3y=-4%29



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




So let's isolate y in the first equation

4x-3y=2 Start with the first equation


-3y=2-4x Subtract 4x from both sides


-3y=-4x%2B2 Rearrange the equation


y=%28-4x%2B2%29%2F%28-3%29 Divide both sides by -3


y=%28%28-4%29%2F%28-3%29%29x%2B%282%29%2F%28-3%29 Break up the fraction


y=%284%2F3%29x-2%2F3 Reduce



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Since y=%284%2F3%29x-2%2F3, we can now replace each y in the second equation with %284%2F3%29x-2%2F3 to solve for x



-2x%2B3highlight%28%28%284%2F3%29x-2%2F3%29%29=-4 Plug in y=%284%2F3%29x-2%2F3 into the second equation. In other words, replace each y with %284%2F3%29x-2%2F3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



-2x%2B%283%29%284%2F3%29x%2B%283%29%28-2%2F3%29=-4 Distribute 3 to %284%2F3%29x-2%2F3


-2x%2B%2812%2F3%29x-6%2F3=-4 Multiply


%283%29%28-2x%2B%2812%2F3%29x-6%2F3%29=%283%29%28-4%29 Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



-6x%2B12x-6=-12 Distribute and multiply the LCM to each side



6x-6=-12 Combine like terms on the left side


6x=-12%2B6Add 6 to both sides


6x=-6 Combine like terms on the right side


x=%28-6%29%2F%286%29 Divide both sides by 6 to isolate x



x=-1 Divide





-----------------First Answer------------------------------


So the first part of our answer is: x=-1









Since we know that x=-1 we can plug it into the equation y=%284%2F3%29x-2%2F3 (remember we previously solved for y in the first equation).



y=%284%2F3%29x-2%2F3 Start with the equation where y was previously isolated.


y=%284%2F3%29%28-1%29-2%2F3 Plug in x=-1


y=-4%2F3-2%2F3 Multiply


y=-2 Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is: y=-2









-----------------Summary------------------------------

So our answers are:

x=-1 and y=-2

which form the point






c)




Start with the given system of equations:
system%284x-3y=2%2C-2x%2B3y=-4%29


Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%284x-3y%29%2B%28-2x%2B3y%29=%282%29%2B%28-4%29


%284x%2B-2x%29%2B%28-3y%2B3y%29=2%2B-4 Group like terms.


2x%2B0y=-2 Combine like terms. Notice how the y terms cancel out.


2x=-2 Simplify.


x=%28-2%29%2F%282%29 Divide both sides by 2 to isolate x.


x=-1 Reduce.


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4x-3y=2 Now go back to the first equation.


4%28-1%29-3y=2 Plug in x=-1.


-4-3y=2 Multiply.


-3y=2%2B4 Add 4 to both sides.


-3y=6 Combine like terms on the right side.


y=%286%29%2F%28-3%29 Divide both sides by -3 to isolate y.


y=-2 Reduce.


So our answer is x=-1 and y=-2.


Which form the ordered pair .


This means that the system is consistent and independent.



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So all of these methods give us the same answer. It's really up to you to determine which one suits you.