Question 394381


Start with the given system of equations:

{{{system(8x+10y=92,24x+11y=162)}}}



{{{-3(8x+10y)=-3(92)}}} Multiply the both sides of the first equation by -3.



{{{-24x-30y=-276}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-24x-30y=-276,24x+11y=162)}}}



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



{{{(-24x-30y)+(24x+11y)=(-276)+(162)}}}



{{{(-24x+24x)+(-30y+11y)=-276+162}}} Group like terms.



{{{0x+-19y=-114}}} Combine like terms.



{{{-19y=-114}}} Simplify.



{{{y=(-114)/(-19)}}} Divide both sides by {{{-19}}} to isolate {{{y}}}.



{{{y=6}}} Reduce.



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{{{-24x-30y=-276}}} Now go back to the first equation.



{{{-24x-30(6)=-276}}} Plug in {{{y=6}}}.



{{{-24x-180=-276}}} Multiply.



{{{-24x=-276+180}}} Add {{{180}}} to both sides.



{{{-24x=-96}}} Combine like terms on the right side.



{{{x=(-96)/(-24)}}} Divide both sides by {{{-24}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



So the solutions are {{{x=4}}} and {{{y=6}}}.



Which form the ordered pair *[Tex \LARGE \left(4,6\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(4,6\right)]. So this visually verifies our answer.



{{{drawing(500,500,-6,14,-4,16,
grid(1),
graph(500,500,-6,14,-4,16,(92-8x)/(10),(162-24x)/(11)),
circle(4,6,0.05),
circle(4,6,0.08),
circle(4,6,0.10)
)}}} Graph of {{{8x+10y=92}}} (red) and {{{24x+11y=162}}} (green) 



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