Question 179497


Start with the given system of equations:

{{{system(4x+5y=40,6x+7y=58)}}}



{{{3(4x+5y)=3(40)}}} Multiply the both sides of the first equation by 3.



{{{12x+15y=120}}} Distribute and multiply.



{{{-2(6x+7y)=-2(58)}}} Multiply the both sides of the second equation by -2.



{{{-12x-14y=-116}}} Distribute and multiply.



So we have the new system of equations:

{{{system(12x+15y=120,-12x-14y=-116)}}}



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+15y)+(-12x-14y)=(120)+(-116)}}}



{{{(12x+-12x)+(15y+-14y)=120+-116}}} Group like terms.



{{{0x+y=4}}} Combine like terms.



{{{y=4}}} Simplify.



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{{{12x+15y=120}}} Now go back to the first equation.



{{{12x+15(4)=120}}} Plug in {{{y=4}}}.



{{{12x+60=120}}} Multiply.



{{{12x=120-60}}} Subtract {{{60}}} from both sides.



{{{12x=60}}} Combine like terms on the right side.



{{{x=(60)/(12)}}} Divide both sides by {{{12}}} to isolate {{{x}}}.



{{{x=5}}} Reduce.



So our answer is {{{x=5}}} and {{{y=4}}}.



Which form the ordered pair *[Tex \LARGE \left(5,4\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(5,4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-6,14,
grid(1),
graph(500,500,-5,15,-6,14,(40-4x)/(5),(58-6x)/(7)),
circle(5,4,0.05),
circle(5,4,0.08),
circle(5,4,0.10)
)}}} Graph of {{{4x+5y=40}}} (red) and {{{6x+7y=58}}} (green)