Question 159294
Start with the given system


{{{y=2x+6}}}
{{{2y+3x=5}}}




{{{2(2x+6)+3x=5}}}  Plug in {{{y=2x+6}}} into the second equation. In other words, replace each {{{y}}} with {{{2x+6}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{4x+12+3x=5}}} Distribute



{{{7x+12=5}}} Combine like terms on the left side



{{{7x=5-12}}}Subtract 12 from both sides



{{{7x=-7}}} Combine like terms on the right side



{{{x=(-7)/(7)}}} Divide both sides by 7 to isolate x




{{{x=-1}}} Divide





Now that we know that {{{x=-1}}}, we can plug this into {{{y=2x+6}}} to find {{{y}}}




{{{y=2(-1)+6}}} Substitute {{{-1}}} for each {{{x}}}



{{{y=-2+6}}} Multiply



{{{y=4}}} Add



So our answer is {{{x=-1}}} and {{{y=4}}} which forms the ordered pair *[Tex \LARGE \left(-1,4\right)]




Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(-1,4\right)]. So this visually verifies our answer.



{{{ drawing(500, 500, -5, 5, -5, 5,
grid(1),
graph( 500, 500, -5, 5, -5, 5, (5-3x)/2, 2x+6)
) }}} Graph of {{{2y+3x=5}}} (red) and {{{y=2x+6}}} (green)




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