Question 146961


Start with the given system of equations:



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



In order to graph these equations, we <font size="4"><b>must</b></font> solve for y first.



Let's graph the first equation:



{{{x-2y=10}}} Start with the first equation.



{{{-2y=10-x}}} Subtract {{{x}}} from both sides.



{{{y=(10-x)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=(1/2)x-5}}} Rearrange the terms and simplify.



Now let's graph the equation:



{{{drawing(500,500,-10,10,-10,10,
grid(0),
graph(500,500,-10,10,-10,10,(1/2)x-5)
)}}} Graph of {{{y=(1/2)x-5}}}.



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Now let's graph the second equation:



{{{x+2y=-2}}} Start with the second equation.



{{{2y=-2-x}}} Subtract {{{x}}} from both sides.



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



{{{y=-(1/2)x-1}}} Rearrange the terms and simplify.



Now let's graph the equation:



{{{drawing(500,500,-10,10,-10,10,
grid(0),
graph(500,500,-10,10,-10,10,-(1/2)x-1)
)}}} Graph of {{{y=-(1/2)x-1}}}.



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Now let's graph the two equations together:



{{{drawing(500,500,-10,10,-10,10,
grid(1),
graph(500,500,-10,10,-10,10,(1/2)x-5,-(1/2)x-1)
)}}} Graph of {{{y=(1/2)x-5}}} (red). Graph of {{{y=-(1/2)x-1}}} (green)



From the graph, we can see that the two lines intersect at the point *[Tex \LARGE \left(4,-3\right)]. So the solution to the system of equations is *[Tex \LARGE \left(4,-3\right)]. This tells us that the system of equations is consistent and independent.