Question 173080

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



{{{3(y+3)-2y=4}}}  Plug in {{{x=y+3}}}. In other words, replace each {{{x}}} with {{{y+3}}}. Notice we've eliminated the {{{x}}} variables. So we now have a simple equation with one unknown.



{{{3y+9-2y=4}}} Distribute



{{{y+9=4}}} Combine like terms on the left side



{{{y=4-9}}}Subtract 9 from both sides



{{{y=-5}}} Combine like terms on the right side





Now that we know that {{{y=-5}}}, we can plug this into {{{x=y+3}}} to find {{{x}}}




{{{x=(-5)+3}}} Substitute {{{-5}}} for each {{{y}}}



{{{x=-2}}} Add



So the solutions are {{{x=-2}}} and {{{y=-5}}} which form the ordered pair *[Tex \LARGE \left(-2,-5\right)]



Since the system has a unique solution, this means that the system is consistent and independent.



Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(-2,-5\right)]. So this verifies our answer.

{{{ drawing(500, 500, -5, 5, -8, 2,
 grid(1),
 graph( 500, 500, -5, 5, -8, 2,(4-3x)/(-2), (x-3)/1)
)}}} Graph of {{{3x-2y=4}}} (red) and {{{x=y+3}}} (green)