Question 200985

Start with the given system of equations:

{{{system(3x+2y=7,5x-2y=1)}}}



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



{{{(3x+2y)+(5x-2y)=(7)+(1)}}}



{{{(3x+5x)+(2y-2y)=7+1}}} Group like terms.



{{{8x+0y=8}}} Combine like terms.



{{{8x=8}}} Simplify.



{{{x=(8)/(8)}}} Divide both sides by {{{8}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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{{{3x+2y=7}}} Now go back to the first equation.



{{{3(1)+2y=7}}} Plug in {{{x=1}}}.



{{{3+2y=7}}} Multiply.



{{{2y=7-3}}} Subtract {{{3}}} from both sides.



{{{2y=4}}} Combine like terms on the right side.



{{{y=(4)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.



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



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



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