Question 184023


Start with the given system of equations:

{{{system(5x+2y=8,2x-3y=7)}}}



{{{3(5x+2y)=3(8)}}} Multiply the both sides of the first equation by 3.



{{{15x+6y=24}}} Distribute and multiply.



{{{2(2x-3y)=2(7)}}} Multiply the both sides of the second equation by 2.



{{{4x-6y=14}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x+6y=24,4x-6y=14)}}}



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



{{{(15x+6y)+(4x-6y)=(24)+(14)}}}



{{{(15x+4x)+(6y+-6y)=24+14}}} Group like terms.



{{{19x+0y=38}}} Combine like terms.



{{{19x=38}}} Simplify.



{{{x=(38)/(19)}}} Divide both sides by {{{19}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



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{{{15x+6y=24}}} Now go back to the first equation.



{{{15(2)+6y=24}}} Plug in {{{x=2}}}.



{{{30+6y=24}}} Multiply.



{{{6y=24-30}}} Subtract {{{30}}} from both sides.



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



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



{{{y=-1}}} Reduce.



So the solutions are {{{x=2}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(2,-1\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(2,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-11,9,
grid(1),
graph(500,500,-8,12,-11,9,(8-5x)/(2),(7-2x)/(-3)),
circle(2,-1,0.05),
circle(2,-1,0.08),
circle(2,-1,0.10)
)}}} Graph of {{{5x+2y=8}}} (red) and {{{2x-3y=7}}} (green)