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

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)   (Show Source): You can put this solution on YOUR website!

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

Lets start with the given system of linear equations

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

Subtract from both sides

Divide both sides by 3.

Which breaks down and reduces to

Now we've fully isolated y

Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x.

Replace y with . Since this eliminates y, we can now solve for x.

Distribute 5 to

Multiply

Reduce any fractions

Subtract from both sides

Combine the terms on the right side

Make 6 into a fraction with a denominator of 3

Now combine the terms on the left side.

Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get

<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Subtract from both sides

Make 180 into a fraction with a denominator of 2

Combine the terms on the right side

Multiply both sides by . This will cancel out 5 on the left side.

Multiply the terms on the right side

Reduce

So this is the other answer

<---------------------------------Other answer

So our solution is

and

which can also look like

(,)

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

we get

graph of (red) and (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 (,). This verifies our answer.

-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Since the solution (,) satisfies the system of equations

this verifies our answer.

Answer by ikleyn(52800)   (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= = 18.75; y = = 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)   (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 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.
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