Question 180714

Start with the given system of equations:

{{{system(4x+10y=2,3x+5y=5)}}}



{{{-2(3x+5y)=-2(5)}}} Multiply the both sides of the second equation by -2.



{{{-6x-10y=-10}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x+10y=2,-6x-10y=-10)}}}



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



{{{(4x+10y)+(-6x-10y)=(2)+(-10)}}}



{{{(4x+-6x)+(10y+-10y)=2+-10}}} Group like terms.



{{{-2x+0y=-8}}} Combine like terms.



{{{-2x=-8}}} Simplify.



{{{x=(-8)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



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{{{4x+10y=2}}} Now go back to the first equation.



{{{4(4)+10y=2}}} Plug in {{{x=4}}}.



{{{16+10y=2}}} Multiply.



{{{10y=2-16}}} Subtract {{{16}}} from both sides.



{{{10y=-14}}} Combine like terms on the right side.



{{{y=(-14)/(10)}}} Divide both sides by {{{10}}} to isolate {{{y}}}.



{{{y=-7/5}}} Reduce.



So the solutions are {{{x=4}}} and {{{y=-7/5}}}



Which form the ordered pair *[Tex \LARGE \left(4,-\frac{7}{5}\right)].



This means that the system is consistent and independent.