SOLUTION: Multiply both sides of each equation by a common denominator to eliminate the fractions. then solve the system x/2 + y/3 = 8 2x/3 + 3y/2 = 17

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Multiply both sides of each equation by a common denominator to eliminate the fractions. then solve the system x/2 + y/3 = 8 2x/3 + 3y/2 = 17      Log On


   

Question 104573: Multiply both sides of each equation by a common denominator to eliminate the fractions. then solve the system
x/2 + y/3 = 8
2x/3 + 3y/2 = 17
Found 2 solutions by jim_thompson5910, stanbon:
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=8 Start with the first equation


6%28%281%2F2%29%2Ax%2B%281%2F3%29%2Ay%29=%286%29%2A%288%29 Multiply both sides by the LCD 6



3%2Ax%2B2%2Ay=48Distribute and simplify


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%282%2F3%29%2Ax%2B%283%2F2%29%2Ay=17 Start with the second equation


6%28%282%2F3%29%2Ax%2B%283%2F2%29%2Ay%29=%286%29%2A%2817%29 Multiply both sides by the LCD 6



4%2Ax%2B9%2Ay=102 Distribute and simplify


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Lets start with the given system of linear equations

3%2Ax%2B2%2Ay=48
4%2Ax%2B9%2Ay=102

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=48-3%2AxSubtract 3%2Ax from both sides

y=%2848-3%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=24-%283%2F2%29%2Ax Now we've fully isolated y

Since y equals 24-%283%2F2%29%2Ax we can substitute the expression 24-%283%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B9%2Ahighlight%28%2824-%283%2F2%29%2Ax%29%29=102 Replace y with 24-%283%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax%2B9%2A%2824%29%2B9%28-3%2F2%29x=102 Distribute 9 to 24-%283%2F2%29%2Ax

4%2Ax%2B216-%2827%2F2%29%2Ax=102 Multiply



4%2Ax%2B216-%2827%2F2%29%2Ax=102 Reduce any fractions

4%2Ax-%2827%2F2%29%2Ax=102-216 Subtract 216 from both sides


4%2Ax-%2827%2F2%29%2Ax=-114 Combine the terms on the right side



%288%2F2%29%2Ax-%2827%2F2%29x=-114 Make 4 into a fraction with a denominator of 2

%28-19%2F2%29%2Ax=-114 Now combine the terms on the left side.


cross%28%282%2F-19%29%28-19%2F2%29%29x=%28-114%2F1%29%282%2F-19%29 Multiply both sides by 2%2F-19. This will cancel out -19%2F2 and isolate x

So when we multiply -114%2F1 and 2%2F-19 (and simplify) we get



x=12 <---------------------------------One answer

Now that we know that x=12, lets substitute that in for x to solve for y

4%2812%29%2B9%2Ay=102 Plug in x=12 into the 2nd equation

48%2B9%2Ay=102 Multiply

9%2Ay=102-48Subtract 48 from both sides

9%2Ay=54 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ay=%2854%2F1%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.

y=54%2F9 Multiply the terms on the right side


y=6 Reduce


So this is the other answer


y=6<---------------------------------Other answer


So our solution is

x=12 and y=6

which can also look like

(12,6)

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

3%2Ax%2B2%2Ay=48
4%2Ax%2B9%2Ay=102

we get


graph of 3%2Ax%2B2%2Ay=48 (red) and 4%2Ax%2B9%2Ay=102 (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 (12,6). This verifies our answer.


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Check:

Plug in (12,6) into the system of equations


Let x=12 and y=6. Now plug those values into the equation 3%2Ax%2B2%2Ay=48

3%2A%2812%29%2B2%2A%286%29=48 Plug in x=12 and y=6


36%2B12=48 Multiply


48=48 Add


48=48 Reduce. Since this equation is true the solution works.


So the solution (12,6) satisfies 3%2Ax%2B2%2Ay=48



Let x=12 and y=6. Now plug those values into the equation 4%2Ax%2B9%2Ay=102

4%2A%2812%29%2B9%2A%286%29=102 Plug in x=12 and y=6


48%2B54=102 Multiply


102=102 Add


102=102 Reduce. Since this equation is true the solution works.


So the solution (12,6) satisfies 4%2Ax%2B9%2Ay=102


Since the solution (12,6) satisfies the system of equations


3%2Ax%2B2%2Ay=48
4%2Ax%2B9%2Ay=102


this verifies our answer.

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
solve the system
x/2 + y/3 = 8
Multiply thru by by 6 to get:
1st: 3x + 2y = 48
----------------
2x/3 + 3y/2 = 17
Multiply thru by 6 to get:
2nd: 4x + 9y = 102
------------------
Multiply 1st by 4 and 2nd by 3 to get:
3rd: 12x + 8y = 192
4th: 12x +27y = 306
-----------------------
Subtract 3rd from 4th to solve for y:
19y = 114
y = 6
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Substitute into 3x+2y=48 to solve for "x".
3x + 12 = 48
3x = 36
x = 12
---------------
Final solution:
x=12; y=6
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Cheers,
Stan H.