Question 215093


Start with the given system of equations:

{{{system(7x+4y=15,4x-4y=-48)}}}



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



{{{(7x+4y)+(4x-4y)=(15)+(-48)}}}



{{{(7x+4x)+(4y+-4y)=15+-48}}} Group like terms.



{{{11x+0y=-33}}} Combine like terms.



{{{11x=-33}}} Simplify.



{{{x=(-33)/(11)}}} Divide both sides by {{{11}}} to isolate {{{x}}}.



{{{x=-3}}} Reduce.



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{{{7x+4y=15}}} Now go back to the first equation.



{{{7(-3)+4y=15}}} Plug in {{{x=-3}}}.



{{{-21+4y=15}}} Multiply.



{{{4y=15+21}}} Add {{{21}}} to both sides.



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



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



{{{y=9}}} Reduce.



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



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



{{{drawing(500,500,-13,7,-1,19,
grid(1),
graph(500,500,-13,7,-1,19,(15-7x)/(4),(-48-4x)/(-4)),
circle(-3,9,0.05),
circle(-3,9,0.08),
circle(-3,9,0.10)
)}}} Graph of {{{7x+4y=15}}} (red) and {{{4x-4y=-48}}} (green)