Question 550512


Start with the given system of equations:

{{{system(-2x-8y=-42,-4x+6y=-18)}}}



{{{-2(-2x-8y)=-2(-42)}}} Multiply the both sides of the first equation by -2.



{{{4x+16y=84}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x+16y=84,-4x+6y=-18)}}}



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+16y)+(-4x+6y)=(84)+(-18)}}}



{{{(4x+-4x)+(16y+6y)=84+-18}}} Group like terms.



{{{0x+22y=66}}} Combine like terms.



{{{22y=66}}} Simplify.



{{{y=(66)/(22)}}} Divide both sides by {{{22}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



------------------------------------------------------------------



{{{4x+16y=84}}} Now go back to the first equation.



{{{4x+16(3)=84}}} Plug in {{{y=3}}}.



{{{4x+48=84}}} Multiply.



{{{4x=84-48}}} Subtract {{{48}}} from both sides.



{{{4x=36}}} Combine like terms on the right side.



{{{x=(36)/(4)}}} Divide both sides by {{{4}}} to isolate {{{x}}}.



{{{x=9}}} Reduce.



So the solutions are {{{x=9}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(9,3\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(9,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-1,19,-7,13,
grid(1),
graph(500,500,-1,19,-7,13,(-42+2x)/(-8),(-18+4x)/(6)),
circle(9,3,0.05),
circle(9,3,0.08),
circle(9,3,0.10)
)}}} Graph of {{{-2x-8y=-42}}} (red) and {{{-4x+6y=-18}}} (green)