Question 168738
# 1

Your first equation has two "x" variables. Is there a "y" term in the first equation?


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# 2



Start with the given system of equations:

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



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



{{{-10x+5y=35}}} Distribute and multiply.



So we have the new system of equations:

{{{system(3x-5y=7,-10x+5y=35)}}}



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



{{{(3x-5y)+(-10x+5y)=(7)+(35)}}}



{{{(3x+-10x)+(-5y+5y)=7+35}}} Group like terms.



{{{-7x+0y=42}}} Combine like terms. Notice how the y terms cancel out.



{{{-7x=42}}} Simplify.



{{{x=(42)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{x}}}.



{{{x=-6}}} Reduce.



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{{{3x-5y=7}}} Now go back to the first equation.



{{{3(-6)-5y=7}}} Plug in {{{x=-6}}}.



{{{-18-5y=7}}} Multiply.



{{{-5y=7+18}}} Add {{{18}}} to both sides.



{{{-5y=25}}} Combine like terms on the right side.



{{{y=(25)/(-5)}}} Divide both sides by {{{-5}}} to isolate {{{y}}}.



{{{y=-5}}} Reduce.



So our answer is {{{x=-6}}} and {{{y=-5}}}.



Which form the ordered pair *[Tex \LARGE \left(-6,-5\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(-6,-5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-16,4,-15,5,
grid(1),
graph(500,500,-16,4,-15,5,(7-3x)/(-5),(-7-2x)/(-1)),
circle(-6,-5,0.05),
circle(-6,-5,0.08),
circle(-6,-5,0.10)
)}}} Graph of {{{3x-5y=7}}} (red) and {{{2x-y=-7}}} (green) 



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# 3




Start with the given system of equations:

{{{system(x-3y=1,3x-5y=-5)}}}



{{{-3(x-3y)=-3(1)}}} Multiply the both sides of the first equation by -3.



{{{-3x+9y=-3}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-3x+9y=-3,3x-5y=-5)}}}



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



{{{(-3x+9y)+(3x-5y)=(-3)+(-5)}}}



{{{(-3x+3x)+(9y+-5y)=-3+-5}}} Group like terms.



{{{0x+4y=-8}}} Combine like terms. Notice how the x terms cancel out.



{{{4y=-8}}} Simplify.



{{{y=(-8)/(4)}}} Divide both sides by {{{4}}} to isolate {{{y}}}.



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



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{{{-3x+9y=-3}}} Now go back to the first equation.



{{{-3x+9(-2)=-3}}} Plug in {{{y=-2}}}.



{{{-3x-18=-3}}} Multiply.



{{{-3x=-3+18}}} Add {{{18}}} to both sides.



{{{-3x=15}}} Combine like terms on the right side.



{{{x=(15)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{x}}}.



{{{x=-5}}} Reduce.



So our answer is {{{x=-5}}} and {{{y=-2}}}.



Which form the ordered pair *[Tex \LARGE \left(-5,-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(-5,-2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-15,5,-12,8,
grid(1),
graph(500,500,-15,5,-12,8,(1-x)/(-3),(-5-3x)/(-5)),
circle(-5,-2,0.05),
circle(-5,-2,0.08),
circle(-5,-2,0.10)
)}}} Graph of {{{x-3y=1}}} (red) and {{{3x-5y=-5}}} (green)