Question 130247
# 1


Start with the given system

{{{x+2y=7}}}
{{{x=y+1}}}




{{{y+1+2y=7}}}  Plug in {{{x=y+1}}} into the first equation. In other words, replace each {{{x}}} with {{{y+1}}}. Notice we've eliminated the {{{x}}} variables. So we now have a simple equation with one unknown.





{{{3y+1=7}}} Combine like terms on the left side



{{{3y=7-1}}}Subtract 1 from both sides



{{{3y=6}}} Combine like terms on the right side



{{{y=(6)/(3)}}} Divide both sides by 3 to isolate y




{{{y=2}}} Divide





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




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



{{{x=3}}} 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, (7-x)/(2), x-1) }}} Graph of {{{x+2y=7}}} (red) and {{{x=y+1}}} (green)





<hr>



# 2



Start with the given system

{{{3x-y=2}}}
{{{y=2x-9}}}




{{{3x-(2x-9)=2}}}  Plug in {{{y=2x-9}}} into the first equation. In other words, replace each {{{y}}} with {{{2x-9}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{3x-2x+9=2}}} Distribute the negative



{{{x+9=2}}} Combine like terms on the left side



{{{x=2-9}}}Subtract 9 from both sides



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





Now that we know that {{{x=-7}}}, we can plug this into {{{y=2x-9}}} to find {{{y}}}




{{{y=2(-7)-9}}} Substitute {{{-7}}} for each {{{x}}}



{{{y=-23}}} Simplify



So our answer is {{{x=-7}}} and {{{y=-23}}} which also looks like *[Tex \LARGE \left(-7,-23\right)]