Question 147952


Start with the given system of equations:



{{{system(3x+y=16,4x+5y=14)}}}



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=16}}} Start with the first equation.



{{{y=16-3x}}} Subtract {{{3x}}} from both sides.



{{{y=-3x+16}}} 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+16)
)}}} Graph of {{{y=-3x+16}}}.



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Now let's graph the second equation:



{{{4x+5y=14}}} Start with the second equation.



{{{5y=14-4x}}} Subtract {{{4x}}} from both sides.



{{{y=(14-4x)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=-(4/5)x+14/5}}} 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,-(4/5)x+14/5)
)}}} Graph of {{{y=-(4/5)x+14/5}}}.



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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+16,-(4/5)x+14/5)
)}}} Graph of {{{y=-3x+16}}} (red). Graph of {{{y=-(4/5)x+14/5}}} (green)



From the graph, we can see that the two lines intersect at the point *[Tex \LARGE \left(6,-2\right)]. So the solution to the system of equations is *[Tex \LARGE \left(6,-2\right)]. This tells us that the system of equations is consistent and independent.