Question 198956


Start with the given system of equations:

{{{system(3x-5y=-9,5x-6y=-8)}}}



{{{5(3x-5y)=5(-9)}}} Multiply the both sides of the first equation by 5.



{{{15x-25y=-45}}} Distribute and multiply.



{{{-3(5x-6y)=-3(-8)}}} Multiply the both sides of the second equation by -3.



{{{-15x+18y=24}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x-25y=-45,-15x+18y=24)}}}



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



{{{(15x-25y)+(-15x+18y)=(-45)+(24)}}}



{{{(15x+-15x)+(-25y+18y)=-45+24}}} Group like terms.



{{{0x+-7y=-21}}} Combine like terms.



{{{-7y=-21}}} Simplify.



{{{y=(-21)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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{{{15x-25y=-45}}} Now go back to the first equation.



{{{15x-25(3)=-45}}} Plug in {{{y=3}}}.



{{{15x-75=-45}}} Multiply.



{{{15x=-45+75}}} Add {{{75}}} to both sides.



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



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



{{{x=2}}} Reduce.



So the solutions are {{{x=2}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(2,3\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(2,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-7,13,
grid(1),
graph(500,500,-8,12,-7,13,(-9-3x)/(-5),(-8-5x)/(-6)),
circle(2,3,0.05),
circle(2,3,0.08),
circle(2,3,0.10)
)}}} Graph of {{{3x-5y=-9}}} (red) and {{{5x-6y=-8}}} (green)