Question 580217
Start with the given system of equations:

{{{system(x+3y=23,-x+4y=12)}}}



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



{{{(x+3y)+(-x+4y)=(23)+(12)}}}



{{{(x-x)+(3y+4y)=23+12}}} Group like terms.



{{{0x+7y=35}}} Combine like terms.



{{{7y=35}}} Simplify.



{{{y=(35)/(7)}}} Divide both sides by {{{7}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{x+3y=23}}} Now go back to the first equation.



{{{x+3(5)=23}}} Plug in {{{y=5}}}.



{{{x+15=23}}} Multiply.



{{{x=23-15}}} Subtract {{{15}}} from both sides.



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



So the solutions are {{{x=8}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(8,5\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(8,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-2,18,-5,15,
grid(1),
graph(500,500,-2,18,-5,15,(23-x)/(3),(12+x)/(4)),
circle(8,5,0.05),
circle(8,5,0.08),
circle(8,5,0.10)
)}}} Graph of {{{x+3y=23}}} (red) and {{{-x+4y=12}}} (green)