Question 376372


Start with the given system of equations:

{{{system(-5x-2y=13,7x-3y=5)}}}



{{{7(-5x-2y)=7(13)}}} Multiply the both sides of the first equation by 7.



{{{-35x-14y=91}}} Distribute and multiply.



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



{{{35x-15y=25}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-35x-14y=91,35x-15y=25)}}}



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



{{{(-35x-14y)+(35x-15y)=(91)+(25)}}}



{{{(-35x+35x)+(-14y+-15y)=91+25}}} Group like terms.



{{{0x+-29y=116}}} Combine like terms.



{{{-29y=116}}} Simplify.



{{{y=(116)/(-29)}}} Divide both sides by {{{-29}}} to isolate {{{y}}}.



{{{y=-4}}} Reduce.



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{{{-35x-14y=91}}} Now go back to the first equation.



{{{-35x-14(-4)=91}}} Plug in {{{y=-4}}}.



{{{-35x+56=91}}} Multiply.



{{{-35x=91-56}}} Subtract {{{56}}} from both sides.



{{{-35x=35}}} Combine like terms on the right side.



{{{x=(35)/(-35)}}} Divide both sides by {{{-35}}} to isolate {{{x}}}.



{{{x=-1}}} Reduce.



So the solutions are {{{x=-1}}} and {{{y=-4}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,-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(-1,-4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-14,6,
grid(1),
graph(500,500,-11,9,-14,6,(13+5x)/(-2),(5-7x)/(-3)),
circle(-1,-4,0.05),
circle(-1,-4,0.08),
circle(-1,-4,0.10)
)}}} Graph of {{{-5x-2y=13}}} (red) and {{{7x-3y=5}}} (green)