Question 181961


Start with the given system of equations:

{{{system(6x+7y=-5,4x+3y=-15)}}}



{{{2(6x+7y)=2(-5)}}} Multiply the both sides of the first equation by 2.



{{{12x+14y=-10}}} Distribute and multiply.



{{{-3(4x+3y)=-3(-15)}}} Multiply the both sides of the second equation by -3.



{{{-12x-9y=45}}} Distribute and multiply.



So we have the new system of equations:

{{{system(12x+14y=-10,-12x-9y=45)}}}



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



{{{(12x+14y)+(-12x-9y)=(-10)+(45)}}}



{{{(12x+-12x)+(14y+-9y)=-10+45}}} Group like terms.



{{{0x+5y=35}}} Combine like terms.



{{{5y=35}}} Simplify.



{{{y=(35)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=7}}} Reduce.



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{{{12x+14y=-10}}} Now go back to the first equation.



{{{12x+14(7)=-10}}} Plug in {{{y=7}}}.



{{{12x+98=-10}}} Multiply.



{{{12x=-10-98}}} Subtract {{{98}}} from both sides.



{{{12x=-108}}} Combine like terms on the right side.



{{{x=(-108)/(12)}}} Divide both sides by {{{12}}} to isolate {{{x}}}.



{{{x=-9}}} Reduce.



So our answer is {{{x=-9}}} and {{{y=7}}}.



Which form the ordered pair *[Tex \LARGE \left(-9,7\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(-9,7\right)]. So this visually verifies our answer.



{{{drawing(500,500,-19,1,-3,17,
grid(1),
graph(500,500,-19,1,-3,17,(-5-6x)/(7),(-15-4x)/(3)),
circle(-9,7,0.05),
circle(-9,7,0.08),
circle(-9,7,0.10)
)}}} Graph of {{{6x+7y=-5}}} (red) and {{{4x+3y=-15}}} (green)