Question 149238


Start with the given system of equations:



{{{system(3x+y=5,-2x+3y=4)}}}



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



Let's graph the first equation:



{{{3x+y=5}}} Start with the first equation.



{{{y=5-3x}}} Subtract {{{3x}}} from both sides.



{{{y=-3x+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,-3x+5)
)}}} Graph of {{{y=-3x+5}}}.



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Now let's graph the second equation:



{{{-2x+3y=4}}} Start with the second equation.



{{{3y=4+2x}}} Add {{{2x}}} to both sides.



{{{y=(4+2x)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y=(2/3)x+4/3}}} 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,(2/3)x+4/3)
)}}} Graph of {{{y=(2/3)x+4/3}}}.



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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,-3x+5,(2/3)x+4/3)
)}}} Graph of {{{y=-3x+5}}} (red). Graph of {{{y=(2/3)x+4/3}}} (green)



From the graph, we can see that the two lines intersect at the point *[Tex \LARGE \left(1,2\right)]. So the solution to the system of equations is *[Tex \LARGE \left(1,2\right)]. This tells us that the system of equations is consistent and independent.