Question 550363


Start with the given system of equations:

{{{system(5x-3y=8,-5x+y=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)+(-5x+y)=(8)+(4)}}}



{{{(5x+-5x)+(-3y+1y)=8+4}}} Group like terms.



{{{0x+-2y=12}}} Combine like terms.



{{{-2y=12}}} Simplify.



{{{y=(12)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=-6}}} Reduce.



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{{{5x-3y=8}}} Now go back to the first equation.



{{{5x-3(-6)=8}}} Plug in {{{y=-6}}}.



{{{5x+18=8}}} Multiply.



{{{5x=8-18}}} Subtract {{{18}}} from both sides.



{{{5x=-10}}} Combine like terms on the right side.



{{{x=(-10)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



{{{x=-2}}} Reduce.



So the solutions are {{{x=-2}}} and {{{y=-6}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,-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(-2,-6\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-16,4,
grid(1),
graph(500,500,-12,8,-16,4,(8-5x)/(-3),4+5x),
circle(-2,-6,0.05),
circle(-2,-6,0.08),
circle(-2,-6,0.10)
)}}} Graph of {{{5x-3y=8}}} (red) and {{{-5x+y=4}}} (green)