Question 568396


Start with the given system of equations:

{{{system(2x-3y=-27,-3x+2y=23)}}}



{{{2(2x-3y)=2(-27)}}} Multiply the both sides of the first equation by 2.



{{{4x-6y=-54}}} Distribute and multiply.



{{{3(-3x+2y)=3(23)}}} Multiply the both sides of the second equation by 3.



{{{-9x+6y=69}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x-6y=-54,-9x+6y=69)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(4x-6y)+(-9x+6y)=(-54)+(69)}}}



{{{(4x+-9x)+(-6y+6y)=-54+69}}} Group like terms.



{{{-5x+0y=15}}} Combine like terms.



{{{-5x=15}}} Simplify.



{{{x=(15)/(-5)}}} Divide both sides by {{{-5}}} to isolate {{{x}}}.



{{{x=-3}}} Reduce.



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



{{{4x-6y=-54}}} Now go back to the first equation.



{{{4(-3)-6y=-54}}} Plug in {{{x=-3}}}.



{{{-12-6y=-54}}} Multiply.



{{{-6y=-54+12}}} Add {{{12}}} to both sides.



{{{-6y=-42}}} Combine like terms on the right side.



{{{y=(-42)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{y}}}.



{{{y=7}}} Reduce.



So the solutions are {{{x=-3}}} and {{{y=7}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,7\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-3,7\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-3,17,
grid(1),
graph(500,500,-13,7,-3,17,(-27-2x)/(-3),(23+3x)/(2)),
circle(-3,7,0.05),
circle(-3,7,0.08),
circle(-3,7,0.10)
)}}} Graph of {{{2x-3y=-27}}} (red) and {{{-3x+2y=23}}} (green) 



<font color=red>-------------------------------------------------------------------------------------------------</font>

<b><font size=3>If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com">jim_thompson5910@hotmail.com</a>


Also, please consider visiting my website: <a href="http://www.freewebs.com/jimthompson5910/home.html">http://www.freewebs.com/jimthompson5910/home.html</a> and making a donation. Thank you


Jim</font></b>

<font color=red>-------------------------------------------------------------------------------------------------</font>