Question 550485


Start with the given system of equations:

{{{system(x+4y=21,-x+3y=0)}}}



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



{{{(x+4y)+(-1x+3y)=(21)+(0)}}}



{{{(1x+-1x)+(4y+3y)=21+0}}} Group like terms.



{{{0x+7y=21}}} Combine like terms.



{{{7y=21}}} Simplify.



{{{y=(21)/(7)}}} Divide both sides by {{{7}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



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{{{x+4y=21}}} Now go back to the first equation.



{{{x+4(3)=21}}} Plug in {{{y=3}}}.



{{{x+12=21}}} Multiply.



{{{x=21-12}}} Subtract {{{12}}} from both sides.



{{{x=9}}} Combine like terms on the right side.



So the solutions are {{{x=9}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(9,3\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(9,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-1,19,-7,13,
grid(1),
graph(500,500,-1,19,-7,13,(21-x)/(4),(0+x)/(3)),
circle(9,3,0.05),
circle(9,3,0.08),
circle(9,3,0.10)
)}}} Graph of {{{x+4y=21}}} (red) and {{{-x+3y=0}}} (green)