Question 211486


Start with the given system of equations:

{{{system(-4x+5y=-10,5x-4y=8)}}}



{{{5(-4x+5y)=5(-10)}}} Multiply the both sides of the first equation by 5.



{{{-20x+25y=-50}}} Distribute and multiply.



{{{4(5x-4y)=4(8)}}} Multiply the both sides of the second equation by 4.



{{{20x-16y=32}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-20x+25y=-50,20x-16y=32)}}}



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



{{{(-20x+25y)+(20x-16y)=(-50)+(32)}}}



{{{(-20x+20x)+(25y+-16y)=-50+32}}} Group like terms.



{{{0x+9y=-18}}} Combine like terms.



{{{9y=-18}}} Simplify.



{{{y=(-18)/(9)}}} Divide both sides by {{{9}}} to isolate {{{y}}}.



{{{y=-2}}} Reduce.



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{{{-20x+25y=-50}}} Now go back to the first equation.



{{{-20x+25(-2)=-50}}} Plug in {{{y=-2}}}.



{{{-20x-50=-50}}} Multiply.



{{{-20x=-50+50}}} Add {{{50}}} to both sides.



{{{-20x=0}}} Combine like terms on the right side.



{{{x=(0)/(-20)}}} Divide both sides by {{{-20}}} to isolate {{{x}}}.



{{{x=0}}} Reduce.



So the solutions are {{{x=0}}} and {{{y=-2}}}.



Which form the ordered pair *[Tex \LARGE \left(0,-2\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(0,-2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-10,10,-12,8,
grid(1),
graph(500,500,-10,10,-12,8,(-10+4x)/(5),(8-5x)/(-4)),
circle(0,-2,0.05),
circle(0,-2,0.08),
circle(0,-2,0.10)
)}}} Graph of {{{-4x+5y=-10}}} (red) and {{{5x-4y=8}}} (green)