SOLUTION: How would you solve this equation, by either using the substitution method or the elimination method? 3x/4-y/3=7/6 x/2+2y/3=5/3

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: How would you solve this equation, by either using the substitution method or the elimination method? 3x/4-y/3=7/6 x/2+2y/3=5/3      Log On


   

Question 106702: How would you solve this equation, by either using the substitution method or the elimination method?
3x/4-y/3=7/6
x/2+2y/3=5/3
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


%283%2F4%29%2Ax%2B%28-1%2F3%29%2Ay=7%2F6 Start with the first equation


12%28%283%2F4%29%2Ax%2B%28-1%2F3%29%2Ay%29=%2812%29%2A%287%2F6%29 Multiply both sides by the LCD 12



9%2Ax%2B-4%2Ay=14Distribute and simplify


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%281%2F2%29%2Ax%2B%282%2F3%29%2Ay=5%2F3 Start with the second equation


6%28%281%2F2%29%2Ax%2B%282%2F3%29%2Ay%29=%286%29%2A%285%2F3%29 Multiply both sides by the LCD 6



3%2Ax%2B4%2Ay=10 Distribute and simplify


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Lets start with the given system of linear equations

9%2Ax-4%2Ay=14
3%2Ax%2B4%2Ay=10

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

-4%2Ay=14-9%2AxSubtract 9%2Ax from both sides

y=%2814-9%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-7%2F2%2B%289%2F4%29%2Ax Now we've fully isolated y

Since y equals -7%2F2%2B%289%2F4%29%2Ax we can substitute the expression -7%2F2%2B%289%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B4%2Ahighlight%28%28-7%2F2%2B%289%2F4%29%2Ax%29%29=10 Replace y with -7%2F2%2B%289%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B4%2A%28-7%2F2%29%2B4%289%2F4%29x=10 Distribute 4 to -7%2F2%2B%289%2F4%29%2Ax

3%2Ax-28%2F2%2B%2836%2F4%29%2Ax=10 Multiply



3%2Ax-14%2B9%2Ax=10 Reduce any fractions

3%2Ax%2B9%2Ax=10%2B14Add 14 to both sides


3%2Ax%2B9%2Ax=24 Combine the terms on the right side



12%2Ax=24 Now combine the terms on the left side.


cross%28%281%2F12%29%2812%2F1%29%29x=%2824%2F1%29%281%2F12%29 Multiply both sides by 1%2F12. This will cancel out 12%2F1 and isolate x

So when we multiply 24%2F1 and 1%2F12 (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

3%282%29%2B4%2Ay=10 Plug in x=2 into the 2nd equation

6%2B4%2Ay=10 Multiply

4%2Ay=10-6Subtract 6 from both sides

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

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

y=4%2F4 Multiply the terms on the right side


y=1 Reduce


So this is the other answer


y=1<---------------------------------Other answer


So our solution is

x=2 and y=1

which can also look like

(2,1)

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

9%2Ax-4%2Ay=14
3%2Ax%2B4%2Ay=10

we get


graph of 9%2Ax-4%2Ay=14 (red) and 3%2Ax%2B4%2Ay=10 (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 (2,1). This verifies our answer.


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Check:

Plug in (2,1) into the system of equations


Let x=2 and y=1. Now plug those values into the equation 9%2Ax-4%2Ay=14

9%2A%282%29-4%2A%281%29=14 Plug in x=2 and y=1


18-4=14 Multiply


14=14 Add


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


So the solution (2,1) satisfies 9%2Ax-4%2Ay=14



Let x=2 and y=1. Now plug those values into the equation 3%2Ax%2B4%2Ay=10

3%2A%282%29%2B4%2A%281%29=10 Plug in x=2 and y=1


6%2B4=10 Multiply


10=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (2,1) satisfies 3%2Ax%2B4%2Ay=10


Since the solution (2,1) satisfies the system of equations


9%2Ax-4%2Ay=14
3%2Ax%2B4%2Ay=10


this verifies our answer.