Question 146683

Start with the given system of equations:



{{{system(7x+y=12,8x+9y=-57)}}}



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



Let's graph the first equation:



{{{7x+y=12}}} Start with the first equation.



{{{y=12-7x}}} Subtract {{{7x}}} from both sides.



{{{y=-7x+12}}} 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,-7x+12)
)}}} Graph of {{{y=-7x+12}}}.



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Now let's graph the second equation:



{{{8x+9y=-57}}} Start with the second equation.



{{{9y=-57-8x}}} Subtract {{{8x}}} from both sides.



{{{y=(-57-8x)/(9)}}} Divide both sides by {{{9}}} to isolate {{{y}}}.



{{{y=-(8/9)x-19/3}}} 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,-(8/9)x-19/3)
)}}} Graph of {{{y=-(8/9)x-19/3}}}.



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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,-7x+12,-(8/9)x-19/3)
)}}} Graph of {{{y=-7x+12}}} (red). Graph of {{{y=-(8/9)x-19/3}}} (green)



From the graph, we can see that the two lines intersect at the point *[Tex \LARGE \left(3,-9\right)]. So the solution to the system of equations is *[Tex \LARGE \left(3,-9\right)]. This tells us that the system of equations is consistent and independent.