SOLUTION: hi! what do x and y equal in this system of linear equation: 2x+3y=78 and 6x+5y=180

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Question 1130668: hi! what do x and y equal in this system of linear equation: 2x+3y=78 and 6x+5y=180
Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

2x%2B3y=78+
6x%2B5y=180

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax%2B3%2Ay=78
6%2Ax%2B5%2Ay=180

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

3%2Ay=78-2%2AxSubtract 2%2Ax from both sides

y=%2878-2%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=26-%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 26-%282%2F3%29%2Ax we can substitute the expression 26-%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


6%2Ax%2B5%2Ahighlight%28%2826-%282%2F3%29%2Ax%29%29=180 Replace y with 26-%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

6%2Ax%2B5%2A%2826%29%2B5%28-2%2F3%29x=180 Distribute 5 to 26-%282%2F3%29%2Ax

6%2Ax%2B130-%2810%2F3%29%2Ax=180 Multiply



6%2Ax%2B130-%2810%2F3%29%2Ax=180 Reduce any fractions

6%2Ax-%2810%2F3%29%2Ax=180-130 Subtract 130 from both sides


6%2Ax-%2810%2F3%29%2Ax=50 Combine the terms on the right side



%2818%2F3%29%2Ax-%2810%2F3%29x=50 Make 6 into a fraction with a denominator of 3

%288%2F3%29%2Ax=50 Now combine the terms on the left side.


cross%28%283%2F8%29%288%2F3%29%29x=%2850%2F1%29%283%2F8%29 Multiply both sides by 3%2F8. This will cancel out 8%2F3 and isolate x

So when we multiply 50%2F1 and 3%2F8 (and simplify) we get



x=75%2F4 <---------------------------------One answer

Now that we know that x=75%2F4, lets substitute that in for x to solve for y

6%2875%2F4%29%2B5%2Ay=180 Plug in x=75%2F4 into the 2nd equation

225%2F2%2B5%2Ay=180 Multiply

5%2Ay=180-225%2F2Subtract 225%2F2 from both sides

5%2Ay=360%2F2-225%2F2 Make 180 into a fraction with a denominator of 2



5%2Ay=135%2F2 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ay=%28135%2F2%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.

y=135%2F10 Multiply the terms on the right side


y=27%2F2 Reduce


So this is the other answer


y=27%2F2<---------------------------------Other answer


So our solution is

x=75%2F4 and y=27%2F2

which can also look like

(75%2F4,27%2F2)

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

2%2Ax%2B3%2Ay=78
6%2Ax%2B5%2Ay=180

we get


graph of 2%2Ax%2B3%2Ay=78 (red) and 6%2Ax%2B5%2Ay=180 (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 (75%2F4,27%2F2). This verifies our answer.


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Check:

Plug in (75%2F4,27%2F2) into the system of equations


Let x=75%2F4 and y=27%2F2. Now plug those values into the equation 2%2Ax%2B3%2Ay=78

2%2A%2875%2F4%29%2B3%2A%2827%2F2%29=78 Plug in x=75%2F4 and y=27%2F2


150%2F4%2B81%2F2=78 Multiply


312%2F4=78 Add


78=78 Reduce. Since this equation is true the solution works.


So the solution (75%2F4,27%2F2) satisfies 2%2Ax%2B3%2Ay=78



Let x=75%2F4 and y=27%2F2. Now plug those values into the equation 6%2Ax%2B5%2Ay=180

6%2A%2875%2F4%29%2B5%2A%2827%2F2%29=180 Plug in x=75%2F4 and y=27%2F2


450%2F4%2B135%2F2=180 Multiply


720%2F4=180 Add


180=180 Reduce. Since this equation is true the solution works.


So the solution (75%2F4,27%2F2) satisfies 6%2Ax%2B5%2Ay=180


Since the solution (75%2F4,27%2F2) satisfies the system of equations


2%2Ax%2B3%2Ay=78
6%2Ax%2B5%2Ay=180


this verifies our answer.

Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Instead of that nonsensical response from the tutor @MathLover1, i will try to produce something more adequate. 


2x + 3y =  78      (1)
6x + 5y = 180      (2)


Apply the elimination method.  Multiply eq(1) by 3 (both sides).  Keep the equation (2) as is.


6x + 9y = 234      (1')
6x + 5y = 180      (2')


Now subtract eq(2') from eq(1'). the terms " 6x " will cancel each other, and you will get a single equation for only one unknown "y".

    (It is how the Elimination method works)


    9y - 5y = 234 - 180

       4y   =   54

        y   =   54/4  = 13.5.


Then from eq(1)   2x = 78 - 3*13.5 = 37.5  ====>  x = 37.5/2 = 18.75.


Answer.  x= 183%2F4 = 18.75;  y = 131%2F2 = 13.5.

Solved.

------------------------

I am very surprised on how the tutor @LoverMath1 treats these problems on solving equation systems.

By applying this "pluggable solver", she turns / transforms / converts very serious educational task of teaching students
into some unreadable and nonsensical text.


Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

hi! what do x and y equal in this system of linear equation: 2x+3y=78 and 6x+5y=180
This is SO, SO, SIMPLE!

2x + 3y = 78 -------- eq (i)

6x + 5y = 180 ------- eq (ii)

- 6x - 9y = - 234 --- Multiplying eq (i) by - 3 ------ eq (iii)

- 4y = - 54 --------- Adding eqs (iii) & (ii)



Now, substitute 27%2F2 for y in eq (i) or (ii) and find the value of x.

An alternative is to ELIMINATE y in the 2 original equations and you'll be able to find x.