Question 164090


Start with the given system of equations:

{{{system(5x-6y=21,x-2y=5)}}}



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



{{{-5x+10y=-25}}} Distribute and multiply.



So we have the new system of equations:

{{{system(5x-6y=21,-5x+10y=-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:



{{{(5x-6y)+(-5x+10y)=(21)+(-25)}}}



{{{(5x+-5x)+(-6y+10y)=21+-25}}} Group like terms.



{{{0x+4y=-4}}} Combine like terms. Notice how the x terms cancel out.



{{{4y=-4}}} Simplify.



{{{y=(-4)/(4)}}} Divide both sides by {{{4}}} to isolate {{{y}}}.



{{{y=-1}}} Reduce.



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{{{5x-6y=21}}} Now go back to the first equation.



{{{5x-6(-1)=21}}} Plug in {{{y=-1}}}.



{{{5x+6=21}}} Multiply.



{{{5x=21-6}}} Subtract {{{6}}} from both sides.



{{{5x=15}}} Combine like terms on the right side.



{{{x=(15)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



So our answer is {{{x=3}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(3,-1\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(3,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-11,9,
grid(1),
graph(500,500,-7,13,-11,9,(21-5x)/(-6),(5-x)/(-2)),
circle(3,-1,0.05),
circle(3,-1,0.08),
circle(3,-1,0.10)
)}}} Graph of {{{5x-6y=21}}} (red) and {{{x-2y=5}}} (green)