Question 122885
If "The sum of two numbers is 76", then the first equation is {{{x+y=76}}}


Also, if "The second is 8 more than 3 times the first", then the second equation is {{{y=3x+8}}}



So we have the system:




{{{x+y=76}}}
{{{y=3x+8}}}




{{{x+3x+8=76}}}  Plug in {{{y=3x+8}}} into the first equation. In other words, replace each {{{y}}} with {{{3x+8}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{x+3x+8=76}}} Distribute



{{{4x+8=76}}} Combine like terms on the left side



{{{4x=76-8}}}Subtract 8 from both sides



{{{4x=68}}} Combine like terms on the right side



{{{x=(68)/(4)}}} Divide both sides by 4 to isolate x




{{{x=17}}} Divide





Now that we know that {{{x=17}}}, we can plug this into {{{y=3x+8}}} to find {{{y}}}




{{{y=3(17)+8}}} Substitute {{{17}}} for each {{{x}}}



{{{y=59}}} Simplify



So our answer is {{{x=17}}} and {{{y=59}}} which also looks like *[Tex \LARGE \left(17,59\right)]




Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(17,59\right)]. So this verifies our answer.



{{{ graph( 500, 500, -5, 20, -5, 62, 76-x, 3x+8) }}} Graph of {{{x+y=76}}} (red) and {{{y=3x+8}}} (green)