Question 180906

I'll do the first one to get you started


d)




Start with the given system of equations:



{{{system(3x+2y=6,y=4-x)}}}



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+2y=6}}} Start with the first equation.



{{{2y=6-3x}}} Subtract {{{3x}}} from both sides.



{{{y=(6-3x)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



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



Note: let me know if you need help graphing equations


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Now let's graph the second equation {{{y=4-x}}}:


{{{drawing(500,500,-10,10,-10,10,
grid(0),
graph(500,500,-10,10,-10,10,4-x)
)}}} Graph of {{{y=4-x}}}.



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



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