Question 187454


Start with the given system of equations:

{{{system(x+4y=1,2x-3y=-9)}}}



{{{-2(x+4y)=-2(1)}}} Multiply the both sides of the first equation by -2.



{{{-2x-8y=-2}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-2x-8y=-2,2x-3y=-9)}}}



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



{{{(-2x-8y)+(2x-3y)=(-2)+(-9)}}}



{{{(-2x+2x)+(-8y+-3y)=-2+-9}}} Group like terms.



{{{0x+-11y=-11}}} Combine like terms.



{{{-11y=-11}}} Simplify.



{{{y=(-11)/(-11)}}} Divide both sides by {{{-11}}} to isolate {{{y}}}.



{{{y=1}}} Reduce.



------------------------------------------------------------------



{{{-2x-8y=-2}}} Now go back to the first equation.



{{{-2x-8(1)=-2}}} Plug in {{{y=1}}}.



{{{-2x-8=-2}}} Multiply.



{{{-2x=-2+8}}} Add {{{8}}} to both sides.



{{{-2x=6}}} Combine like terms on the right side.



{{{x=(6)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}.



{{{x=-3}}} Reduce.



So the solutions are {{{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,-13,7,-9,11,
grid(1),
graph(500,500,-13,7,-9,11,(1-x)/(4),(-9-2x)/(-3)),
circle(-3,1,0.05),
circle(-3,1,0.08),
circle(-3,1,0.10)
)}}} Graph of {{{x+4y=1}}} (red) and {{{2x-3y=-9}}} (green)