Question 222939


Start with the given system of equations:

{{{system(4x+5y=-41,3x-2y=21)}}}



{{{2(4x+5y)=2(-41)}}} Multiply the both sides of the first equation by 2.



{{{8x+10y=-82}}} Distribute and multiply.



{{{5(3x-2y)=5(21)}}} Multiply the both sides of the second equation by 5.



{{{15x-10y=105}}} Distribute and multiply.



So we have the new system of equations:

{{{system(8x+10y=-82,15x-10y=105)}}}



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



{{{(8x+10y)+(15x-10y)=(-82)+(105)}}}



{{{(8x+15x)+(10y+-10y)=-82+105}}} Group like terms.



{{{23x+0y=23}}} Combine like terms. Notice how the y terms cancel out.



{{{23x=23}}} Simplify.



{{{x=(23)/(23)}}} Divide both sides by {{{23}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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{{{8x+10y=-82}}} Now go back to the first equation.



{{{8(1)+10y=-82}}} Plug in {{{x=1}}}.



{{{8+10y=-82}}} Multiply.



{{{10y=-82-8}}} Subtract {{{8}}} from both sides.



{{{10y=-90}}} Combine like terms on the right side.



{{{y=(-90)/(10)}}} Divide both sides by {{{10}}} to isolate {{{y}}}.



{{{y=-9}}} Reduce.



So our answer is {{{x=1}}} and {{{y=-9}}}.



Which form the ordered pair *[Tex \LARGE \left(1,-9\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(1,-9\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,-19,1,
grid(1),
graph(500,500,-9,11,-19,1,(-41-4x)/(5),(21-3x)/(-2)),
circle(1,-9,0.05),
circle(1,-9,0.08),
circle(1,-9,0.10)
)}}} Graph of {{{4x+5y=-41}}} (red) and {{{3x-2y=21}}} (green)