Question 549803


Start with the given system of equations:

{{{system(10x+15y=20,-9x-15y=-21)}}}



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



{{{(10x+15y)+(-9x-15y)=(20)+(-21)}}}



{{{(10x+-9x)+(15y+-15y)=20+-21}}} Group like terms.



{{{x+0y=-1}}} Combine like terms.



{{{x=-1}}} Simplify.



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{{{10x+15y=20}}} Now go back to the first equation.



{{{10(-1)+15y=20}}} Plug in {{{x=-1}}}.



{{{-10+15y=20}}} Multiply.



{{{15y=20+10}}} Add {{{10}}} to both sides.



{{{15y=30}}} Combine like terms on the right side.



{{{y=(30)/(15)}}} Divide both sides by {{{15}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



So the solutions are {{{x=-1}}} and {{{y=2}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,2\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(-1,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-8,12,
grid(1),
graph(500,500,-11,9,-8,12,(20-10x)/(15),(-21+9x)/(-15)),
circle(-1,2,0.05),
circle(-1,2,0.08),
circle(-1,2,0.10)
)}}} Graph of {{{10x+15y=20}}} (red) and {{{-9x-15y=-21}}} (green)