Question 177388


Start with the given system of equations:

{{{system(10x-8y=-46,x+2y=1)}}}



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



{{{4x+8y=4}}} Distribute and multiply.



So we have the new system of equations:

{{{system(10x-8y=-46,4x+8y=4)}}}



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



{{{(10x-8y)+(4x+8y)=(-46)+(4)}}}



{{{(10x+4x)+(-8y+8y)=-46+4}}} Group like terms.



{{{14x+0y=-42}}} Combine like terms.



{{{14x=-42}}} Simplify.



{{{x=(-42)/(14)}}} Divide both sides by {{{14}}} to isolate {{{x}}}.



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



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{{{10x-8y=-46}}} Now go back to the first equation.



{{{10(-3)-8y=-46}}} Plug in {{{x=-3}}}.



{{{-30-8y=-46}}} Multiply.



{{{-8y=-46+30}}} Add {{{30}}} to both sides.



{{{-8y=-16}}} Combine like terms on the right side.



{{{y=(-16)/(-8)}}} Divide both sides by {{{-8}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



So our answer is {{{x=-3}}} and {{{y=2}}}.



Which form the ordered pair *[Tex \LARGE \left(-3,2\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(-3,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-13,7,-8,12,
grid(1),
graph(500,500,-13,7,-8,12,(-46-10x)/(-8),(1-x)/(2)),
circle(-3,2,0.05),
circle(-3,2,0.08),
circle(-3,2,0.10)
)}}} Graph of {{{10x-8y=-46}}} (red) and {{{x+2y=1}}} (green)