Question 550712


Start with the given system of equations:

{{{system(2x-5y=9,-3x+y=-7)}}}



{{{5(-3x+y)=5(-7)}}} Multiply the both sides of the second equation by 5.



{{{-15x+5y=-35}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x-5y=9,-15x+5y=-35)}}}



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



{{{(2x-5y)+(-15x+5y)=(9)+(-35)}}}



{{{(2x+-15x)+(-5y+5y)=9+-35}}} Group like terms.



{{{-13x+0y=-26}}} Combine like terms.



{{{-13x=-26}}} Simplify.



{{{x=(-26)/(-13)}}} Divide both sides by {{{-13}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



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{{{2x-5y=9}}} Now go back to the first equation.



{{{2(2)-5y=9}}} Plug in {{{x=2}}}.



{{{4-5y=9}}} Multiply.



{{{-5y=9-4}}} Subtract {{{4}}} from both sides.



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



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



{{{y=-1}}} Reduce.



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



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



{{{drawing(500,500,-8,12,-11,9,
grid(1),
graph(500,500,-8,12,-11,9,(9-2x)/(-5),-7+3x),
circle(2,-1,0.05),
circle(2,-1,0.08),
circle(2,-1,0.10)
)}}} Graph of {{{2x-5y=9}}} (red) and {{{-3x+y=-7}}} (green)