Question 197571

Start with the given system of equations:

{{{system(2x-3y=3,5x+2y=17)}}}



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



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



{{{3(5x+2y)=3(17)}}} Multiply the both sides of the second equation by 3.



{{{15x+6y=51}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x-6y=6,15x+6y=51)}}}



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)+(15x+6y)=(6)+(51)}}}



{{{(4x+15x)+(-6y+6y)=6+51}}} Group like terms.



{{{19x+0y=57}}} Combine like terms.



{{{19x=57}}} Simplify.



{{{x=(57)/(19)}}} Divide both sides by {{{19}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



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



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



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



{{{-6y=6-12}}} Subtract {{{12}}} from both sides.



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



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



{{{y=1}}} Reduce.



So the solutions are {{{x=3}}} and {{{y=1}}}.



Which form the ordered pair *[Tex \LARGE \left(3,1\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(3,1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-9,11,
grid(1),
graph(500,500,-7,13,-9,11,(3-2x)/(-3),(17-5x)/(2)),
circle(3,1,0.05),
circle(3,1,0.08),
circle(3,1,0.10)
)}}} Graph of {{{2x-3y=3}}} (red) and {{{5x+2y=17}}} (green)