Question 120449
Start with the given system

{{{-7x-y=1}}}
{{{y=x+3}}}




{{{-7x-(x+3)=1}}}  Plug in {{{y=x+3}}} into the first equation. In other words, replace each {{{y}}} with {{{x+3}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{-7x-x-3=1}}} Distribute the negative



{{{-8x-3=1}}} Combine like terms on the left side



{{{-8x=1+3}}}Add 3 to both sides



{{{-8x=4}}} Combine like terms on the right side



{{{x=(4)/(-8)}}} Divide both sides by -8 to isolate x




{{{x=-1/2}}} Reduce





Now that we know that {{{x=-1/2}}}, we can plug this into {{{y=x+3}}} to find {{{y}}}




{{{y=(-1/2)+3}}} Substitute {{{-1/2}}} for each {{{x}}}



{{{y=5/2}}} Combine like terms



So our answer is {{{x=-1/2}}} and {{{y=5/2}}} which also looks like *[Tex \LARGE \left(-\frac{1}{2},\frac{5}{2}\right)]




Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(-\frac{1}{2},\frac{5}{2}\right)]. So this verifies our answer.



{{{ graph( 500, 500, -5, 5, -5, 5, (1+7x)/-1, x+3) }}} Graph of {{{-7x-y=1}}} (red) and {{{y=x+3}}} (green)