Question 147639


Start with the given system of equations:

{{{system(2x + 7y = -8,-2x + 3y = -12)}}}



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



{{{(2x+7y)+(-2x+3y)=(-8)+(-12)}}}



{{{(2x+-2x)+(7y+3y)=-8+-12}}} Group like terms.



{{{0x+10y=-20}}} Combine like terms. Notice how the x terms cancel out.



{{{10y=-20}}} Simplify.



{{{y=(-20)/(10)}}} Divide both sides by {{{10}}} to isolate {{{y}}}.



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



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{{{2x+7y=-8}}} Now go back to the first equation.



{{{2x+7(-2)=-8}}} Plug in {{{y=-2}}}.



{{{2x-14=-8}}} Multiply.



{{{2x=-8+14}}} Add {{{14}}} 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 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,-7,13,-12,8,
grid(1),
graph(500,500,-7,13,-12,8,(-8-2x)/(7),(-12+2x)/(3)),
circle(3,-2,0.05),
circle(3,-2,0.08),
circle(3,-2,0.10)
)}}} Graph of {{{2x+7y=-8}}} (red) and {{{-2x+3y=-12}}} (green)