Question 252597


Start with the given system of equations:

{{{system(3x+2y=4,4x+3y=7)}}}



{{{3(3x+2y)=3(4)}}} Multiply the both sides of the first equation by 3.



{{{9x+6y=12}}} Distribute and multiply.



{{{-2(4x+3y)=-2(7)}}} Multiply the both sides of the second equation by -2.



{{{-8x-6y=-14}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9x+6y=12,-8x-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:



{{{(9x+6y)+(-8x-6y)=(12)+(-14)}}}



{{{(9x+-8x)+(6y+-6y)=12+-14}}} Group like terms.



{{{x+0y=-2}}} Combine like terms.



{{{x=-2}}} Simplify.



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{{{9x+6y=12}}} Now go back to the first equation.



{{{9(-2)+6y=12}}} Plug in {{{x=-2}}}.



{{{-18+6y=12}}} Multiply.



{{{6y=12+18}}} Add {{{18}}} to both sides.



{{{6y=30}}} Combine like terms on the right side.



{{{y=(30)/(6)}}} Divide both sides by {{{6}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



So the solutions are {{{x=-2}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,5\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(-2,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-5,15,
grid(1),
graph(500,500,-12,8,-5,15,(4-3x)/(2),(7-4x)/(3)),
circle(-2,5,0.05),
circle(-2,5,0.08),
circle(-2,5,0.10)
)}}} Graph of {{{3x+2y=4}}} (red) and {{{4x+3y=7}}} (green)