SOLUTION: Solve by elimination x+3y=-13 2x=3y+10

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Question 1171036: Solve by elimination
x+3y=-13
2x=3y+10
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

x%2B3y=-13
2x=3y%2B10-> write in standard form
-----------------------
x%2B3y=-13
2x-3y=10

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B3%2Ay=-13
2%2Ax-3%2Ay=10

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 2 is 2, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%281%2Ax%2B3%2Ay%29=%28-13%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%282%2Ax-3%2Ay%29=%2810%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
2%2Ax%2B6%2Ay=-26
-2%2Ax%2B3%2Ay=-10

Notice how 2 and -2 add to zero (ie 2%2B-2=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%286%2Ay%2B3%2Ay%29=-26-10

%282-2%29%2Ax%2B%286%2B3%29y=-26-10

cross%282%2B-2%29%2Ax%2B%286%2B3%29%2Ay=-26-10 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

9%2Ay=-36

y=-36%2F9 Divide both sides by 9 to solve for y



y=-4 Reduce


Now plug this answer into the top equation 1%2Ax%2B3%2Ay=-13 to solve for x

1%2Ax%2B3%28-4%29=-13 Plug in y=-4


1%2Ax-12=-13 Multiply



1%2Ax=-13%2B12 Subtract -12 from both sides

1%2Ax=-1 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%28-1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=-1 Multiply the terms on the right side


So our answer is

x=-1, y=-4

which also looks like

(-1, -4)

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

1%2Ax%2B3%2Ay=-13
2%2Ax-3%2Ay=10

we get



graph of 1%2Ax%2B3%2Ay=-13 (red) 2%2Ax-3%2Ay=10 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-1,-4). This verifies our answer.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Let me present you simple and short solution, expressed in clear human language. 


Your starting equations are

    x + 3y = -13       (1)

    2x = 3y + 10       (2)


Keep the first equation as is.
In the second equation, move the term 3y from the right side to the left side, changing its sign.  You will get

     x + 3y = -13      (3)

    2x - 3y =  10      (4)


    +-----------------------------------------------------------+
    |   Now the system of equations is in its standard form,    |
    |   and we can apply the Elimination method.                |
    +-----------------------------------------------------------+


For it, add equations (3) and (4).  The terms "3y" and "-3y" will cancel each other, and you will get the final equation

    3x       = -3

for one single unknown x.  (It is how the elimination method works).


From this last equation,  x = -3/3 = -1.


Now substitute the found value x= -1 into either of original equations to get y.


I will substitute it into equation (1)

    -1 + 3y = -13


which gives

         3y = -13 + 1 = -12,

and then  y = -12/3 = -4.


The problem is just solved.  The  ANSWER  is  x= -1,  y= -4.


You may check the answer by substituting the found values into the original equations.


    - Did I say   ". . . you may ?"  - - -  No, I mean  " you MUST ".


I leave this check to you.

Solved, answered, and explained.