Question 128165
{{{-2x+3y=6}}}
{{{3x-5y=-11}}}



The substitution method means to take an expression that is equal to one of the variables and replacing that variable with the expression.  To get the needed expression, you need to solve for one of the variables -- doesn't matter which.  A good first step would be to put one of the equations into the {{{y=mx+b}}} form, because that is just the equation solved for y.  I would use your work for this but, unfortunately, you did not complete the process of putting the equation into {{{y=mx+b}}} form, because in one case you have 3y on the left and in the other case you have -5y on the left.  The equation is not in {{{y=mx+b}}} form unless the ONLY thing on the left is y.


So, let's finish solving the first equation for y:
{{{-2x+3y=6 }}}
{{{3y=2x+6}}}
{{{y=2x/3+2}}}


Now we have an expression that we know is equal to y.  We can substitute this expression into the other given equation.


{{{3x-red(5y)=-11}}}
{{{3x-red(5(2x/3+2))=-11}}}


Now distribute the -5:
{{{3x-10x/3-10=-11}}}


Multiply by 3 to get rid of that pesky denominator:
{{{9x-10x-30=-33}}}


Add 30 to both sides and collect like terms:
{{{-x=-3}}}


And finally, multipy by -1:
{{{x=3}}}


Now that we have a value for x, we can substitute back into either one of the original equations to find the value for y.  In fact, we can use the first equation that we solved for y in its {{{y=mx+b}}} form.


{{{y=2x/3+2}}}
{{{y=(2/3)(3)+2}}}
{{{y=2+2}}}
{{{y=4}}}


And the solution set for the system consists of the single ordered pair (3,4).


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Elimination method (also known as the addition method):  The idea here is to eliminate one of the variables by first multiplying either one or both of the equation by a constant or constants so that the coefficients on one of the variables become additive inverses (meaning that you get 0 when you add them).


Let's work on elimination of the x term.

{{{-2x+3y=6}}}
{{{3x-5y=-11}}}


If we multiply the first equation by 3, then the coefficient on the x-term will be -6.  Then if we multiply the second equation by 2, the coefficient on that x-term will be 6.  -6 and 6 are additive inverses, i.e. their sum is 0.


{{{-6x+9y=18}}}
{{{6x-10y=-22}}}


Now, add the equations term-by-term.  That is, add the x terms together, then the y terms and then the constants, resulting in:


{{{0x-y=-4}}}


Multiply by -1:
{{{y=4}}}


Now you can do one of two things.  You can take this value for y and substitute it back into either of the original equations and solve for x, or you can go back to the original two equations and repeat the elimination process except that you would multiply the first equation by 5 and the second equation by 3 giving you y term coefficients of 15 and -15.  Either way, you will get a value of 3 for x.  I'll leave it to you to work it out and verify the answer.  The result is the same:  The solution set for the system is the ordered pair (3,4).


I sincerely hope that you are not surprised that the two methods produced the same result.


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Graphing.


By now you should know how to graph the equation of a straight line, so I'll just skip over those details.  To solve a system of equations by graphing, you simply need to sketch a graph of both equations and determine their point of intersection.


{{{drawing(600,600,-1,5,-1,5,
grid(1),
graph(600,600,-1,5,-1,5,2x/3+2,3x/5+11/5),
locate(3.05,3.95,S(3,4)),
circle(3,4,.05)

)}}}