Question 550736


Start with the given system of equations:

{{{system(-8x-2y=-28,-3x+5y=47)}}}



{{{3(-8x-2y)=3(-28)}}} Multiply the both sides of the first equation by 3.



{{{-24x-6y=-84}}} Distribute and multiply.



{{{-8(-3x+5y)=-8(47)}}} Multiply the both sides of the second equation by -8.



{{{24x-40y=-376}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-24x-6y=-84,24x-40y=-376)}}}



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-6y)+(24x-40y)=(-84)+(-376)}}}



{{{(-24x+24x)+(-6y+-40y)=-84+-376}}} Group like terms.



{{{0x+-46y=-460}}} Combine like terms.



{{{-46y=-460}}} Simplify.



{{{y=(-460)/(-46)}}} Divide both sides by {{{-46}}} to isolate {{{y}}}.



{{{y=10}}} Reduce.



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{{{-24x-6y=-84}}} Now go back to the first equation.



{{{-24x-6(10)=-84}}} Plug in {{{y=10}}}.



{{{-24x-60=-84}}} Multiply.



{{{-24x=-84+60}}} Add {{{60}}} to both sides.



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



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



{{{x=1}}} Reduce.



So the solutions are {{{x=1}}} and {{{y=10}}}.



Which form the ordered pair *[Tex \LARGE \left(1,10\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(1,10\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,0,20,
grid(1),
graph(500,500,-9,11,0,20,(-28+8x)/(-2),(47+3x)/(5)),
circle(1,10,0.05),
circle(1,10,0.08),
circle(1,10,0.10)
)}}} Graph of {{{-8x-2y=-28}}} (red) and {{{-3x+5y=47}}} (green)