Question 183269


Start with the given system of equations:

{{{system(5x-3y=32,4x+3y=4)}}}



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



{{{(5x-3y)+(4x+3y)=(32)+(4)}}}



{{{(5x+4x)+(-3y+3y)=32+4}}} Group like terms.



{{{9x+0y=36}}} Combine like terms.



{{{9x=36}}} Simplify.



{{{x=(36)/(9)}}} Divide both sides by {{{9}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



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{{{5x-3y=32}}} Now go back to the first equation.



{{{5(4)-3y=32}}} Plug in {{{x=4}}}.



{{{20-3y=32}}} Multiply.



{{{-3y=32-20}}} Subtract {{{20}}} from both sides.



{{{-3y=12}}} Combine like terms on the right side.



{{{y=(12)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}.



{{{y=-4}}} Reduce.



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



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



{{{drawing(500,500,-6,14,-14,6,
grid(1),
graph(500,500,-6,14,-14,6,(32-5x)/(-3),(4-4x)/(3)),
circle(4,-4,0.05),
circle(4,-4,0.08),
circle(4,-4,0.10)
)}}} Graph of {{{5x-3y=32}}} (red) and {{{4x+3y=4}}} (green)