SOLUTION: Solve the system 2/3x-2/5y=2 0.03x+0.04y=0.67

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Question 475893: Solve the system
2/3x-2/5y=2
0.03x+0.04y=0.67
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


%282%2F3%29%2Ax%2B%28-2%2F5%29%2Ay=2 Start with the first equation


15%28%282%2F3%29%2Ax%2B%28-2%2F5%29%2Ay%29=%2815%29%2A%282%29 Multiply both sides by the LCD 15



10%2Ax%2B-6%2Ay=30Distribute and simplify


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%283%2F100%29%2Ax%2B%284%2F100%29%2Ay=67%2F100 Start with the second equation


100%28%283%2F100%29%2Ax%2B%284%2F100%29%2Ay%29=%28100%29%2A%2867%2F100%29 Multiply both sides by the LCD 100



3%2Ax%2B4%2Ay=67 Distribute and simplify


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Lets start with the given system of linear equations

10%2Ax-6%2Ay=30
3%2Ax%2B4%2Ay=67

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

-6%2Ay=30-10%2AxSubtract 10%2Ax from both sides

y=%2830-10%2Ax%29%2F-6 Divide both sides by -6.


Which breaks down and reduces to



y=-5%2B%285%2F3%29%2Ax Now we've fully isolated y

Since y equals -5%2B%285%2F3%29%2Ax we can substitute the expression -5%2B%285%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B4%2Ahighlight%28%28-5%2B%285%2F3%29%2Ax%29%29=67 Replace y with -5%2B%285%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B4%2A%28-5%29%2B4%285%2F3%29x=67 Distribute 4 to -5%2B%285%2F3%29%2Ax

3%2Ax-20%2B%2820%2F3%29%2Ax=67 Multiply



3%2Ax-20%2B%2820%2F3%29%2Ax=67 Reduce any fractions

3%2Ax%2B%2820%2F3%29%2Ax=67%2B20Add 20 to both sides


3%2Ax%2B%2820%2F3%29%2Ax=87 Combine the terms on the right side



%289%2F3%29%2Ax%2B%2820%2F3%29x=87 Make 3 into a fraction with a denominator of 3

%2829%2F3%29%2Ax=87 Now combine the terms on the left side.


cross%28%283%2F29%29%2829%2F3%29%29x=%2887%2F1%29%283%2F29%29 Multiply both sides by 3%2F29. This will cancel out 29%2F3 and isolate x

So when we multiply 87%2F1 and 3%2F29 (and simplify) we get



x=9 <---------------------------------One answer

Now that we know that x=9, lets substitute that in for x to solve for y

3%289%29%2B4%2Ay=67 Plug in x=9 into the 2nd equation

27%2B4%2Ay=67 Multiply

4%2Ay=67-27Subtract 27 from both sides

4%2Ay=40 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%2840%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=40%2F4 Multiply the terms on the right side


y=10 Reduce


So this is the other answer


y=10<---------------------------------Other answer


So our solution is

x=9 and y=10

which can also look like

(9,10)

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

10%2Ax-6%2Ay=30
3%2Ax%2B4%2Ay=67

we get


graph of 10%2Ax-6%2Ay=30 (red) and 3%2Ax%2B4%2Ay=67 (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 (9,10). This verifies our answer.


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Check:

Plug in (9,10) into the system of equations


Let x=9 and y=10. Now plug those values into the equation 10%2Ax-6%2Ay=30

10%2A%289%29-6%2A%2810%29=30 Plug in x=9 and y=10


90-60=30 Multiply


30=30 Add


30=30 Reduce. Since this equation is true the solution works.


So the solution (9,10) satisfies 10%2Ax-6%2Ay=30



Let x=9 and y=10. Now plug those values into the equation 3%2Ax%2B4%2Ay=67

3%2A%289%29%2B4%2A%2810%29=67 Plug in x=9 and y=10


27%2B40=67 Multiply


67=67 Add


67=67 Reduce. Since this equation is true the solution works.


So the solution (9,10) satisfies 3%2Ax%2B4%2Ay=67


Since the solution (9,10) satisfies the system of equations


10%2Ax-6%2Ay=30
3%2Ax%2B4%2Ay=67


this verifies our answer.





or
2%2F3x-2%2F5y=2

plug in the solution (9,10)
%282%2F3%299-%282%2F5%2910=2
%282%2Fcross%283%29%29cross%289%293-%282%2Fcross%285%29%29cross%2810%292=2
2%2A3-2%2A2=2
6-4=2
2=2


0.03x%2B0.04y=0.67
0.03%2A9%2B0.04%2A10=0.67
0.27%2B0.4=0.67
0.67=0.67