Question 187752

Start with the given system of equations:

{{{system(2x+y=7,3x+y=12)}}}



{{{-1(3x+y)=-1(12)}}} Multiply the both sides of the second equation by -1.



{{{-3x-y=-12}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x+y=7,-3x-y=-12)}}}



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+y)+(-3x-y)=(7)+(-12)}}}



{{{(2x-3x)+(y-y)=7-12}}} Group like terms.



{{{-x+0y=-5}}} Combine like terms.



{{{-x=-5}}} Simplify.



{{{x=(-5)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=5}}} Reduce.



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{{{2x+y=7}}} Now go back to the first equation.



{{{2(5)+y=7}}} Plug in {{{x=5}}}.



{{{10+y=7}}} Multiply.



{{{y=7-10}}} Subtract {{{10}}} from both sides.



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



So the solutions are {{{x=5}}} and {{{y=-3}}}.



Which form the ordered pair *[Tex \LARGE \left(5,-3\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,-3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-13,7,
grid(1),
graph(500,500,-5,15,-13,7,7-2x,12-3x),
circle(5,-3,0.05),
circle(5,-3,0.08),
circle(5,-3,0.10)
)}}} Graph of {{{2x+y=7}}} (red) and {{{3x+y=12}}} (green)