Question 174146


Start with the given system of equations:

{{{system(2x-5y=-6,8x+3y=68)}}}



{{{-4(2x-5y)=-4(-6)}}} Multiply the both sides of the first equation by -4.



{{{-8x+20y=24}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-8x+20y=24,8x+3y=68)}}}



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+20y)+(8x+3y)=(24)+(68)}}}



{{{(-8x+8x)+(20y+3y)=24+68}}} Group like terms.



{{{0x+23y=92}}} Combine like terms. Notice how the x terms cancel out.



{{{23y=92}}} Simplify.



{{{y=(92)/(23)}}} Divide both sides by {{{23}}} to isolate {{{y}}}.



{{{y=4}}} Reduce.



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{{{-8x+20y=24}}} Now go back to the first equation.



{{{-8x+20(4)=24}}} Plug in {{{y=4}}}.



{{{-8x+80=24}}} Multiply.



{{{-8x=24-80}}} Subtract {{{80}}} from both sides.



{{{-8x=-56}}} Combine like terms on the right side.



{{{x=(-56)/(-8)}}} Divide both sides by {{{-8}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



So our answer is {{{x=7}}} and {{{y=4}}}.



Which form the ordered pair *[Tex \LARGE \left(7,4\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(7,4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-6,14,
grid(1),
graph(500,500,-3,17,-6,14,(-6-2x)/(-5),(68-8x)/(3)),
circle(7,4,0.05),
circle(7,4,0.08),
circle(7,4,0.10)
)}}} Graph of {{{2x-5y=-6}}} (red) and {{{8x+3y=68}}} (green)