Question 200627

Start with the given system of equations:

{{{system(7m+n=27,m-8n=69)}}}



{{{8(7m+n)=8(27)}}} Multiply the both sides of the first equation by 8.



{{{56m+8n=216}}} Distribute and multiply.



So we have the new system of equations:

{{{system(56m+8n=216,m-8n=69)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(56m+8n)+(m-8n)=(216)+(69)}}}



{{{(56m+m)+(8n-8n)=216+69}}} Group like terms.



{{{57m+0n=285}}} Combine like terms.



{{{57m=285}}} Simplify.



{{{m=(285)/(57)}}} Divide both sides by {{{57}}} to isolate {{{m}}}.



{{{m=5}}} Reduce.



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{{{56m+8n=216}}} Now go back to the first equation.



{{{56(5)+8n=216}}} Plug in {{{m=5}}}.



{{{280+8n=216}}} Multiply.



{{{8n=216-280}}} Subtract {{{280}}} from both sides.



{{{8n=-64}}} Combine like terms on the right side.



{{{n=(-64)/(8)}}} Divide both sides by {{{8}}} to isolate {{{n}}}.



{{{n=-8}}} Reduce.



So the solutions are {{{m=5}}} and {{{n=-8}}}.



This means that the system is consistent and independent.