Question 130971

Start with the given system

{{{2x+3y=0}}}
{{{y=3x+11}}}




{{{2x+3(3x+11)=0}}}  Plug in {{{y=3x+11}}} into the first equation. In other words, replace each {{{y}}} with {{{3x+11}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{2x+9x+33=0}}} Distribute



{{{11x+33=0}}} Combine like terms on the left side



{{{11x=0-33}}}Subtract 33 from both sides



{{{11x=-33}}} Combine like terms on the right side



{{{x=(-33)/(11)}}} Divide both sides by 11 to isolate x




{{{x=-3}}} Divide





Now that we know that {{{x=-3}}}, we can plug this into {{{y=3x+11}}} to find {{{y}}}




{{{y=3(-3)+11}}} Substitute {{{-3}}} for each {{{x}}}



{{{y=2}}} Simplify



So our answer is {{{x=-3}}} and {{{y=2}}} which also looks like *[Tex \LARGE \left(-3,2\right)]




Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(-3,2\right)]. So this verifies our answer.



{{{ graph( 500, 500, -5, 5, -5, 5, -2x/3, 3x+11) }}} Graph of {{{2x+3y=0}}} (red) and {{{y=3x+11}}} (green)