Question 299407

Start with the given system of equations:

{{{system(4x-7y=4,5x+2y=5)}}}



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



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



{{{7(5x+2y)=7(5)}}} Multiply the both sides of the second equation by 7.



{{{35x+14y=35}}} Distribute and multiply.



So we have the new system of equations:

{{{system(8x-14y=8,35x+14y=35)}}}



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



{{{(8x-14y)+(35x+14y)=(8)+(35)}}}



{{{(8x+35x)+(-14y+14y)=8+35}}} Group like terms.



{{{43x+0y=43}}} Combine like terms.



{{{43x=43}}} Simplify.



{{{x=(43)/(43)}}} Divide both sides by {{{43}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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{{{8x-14y=8}}} Now go back to the first equation.



{{{8(1)-14y=8}}} Plug in {{{x=1}}}.



{{{8-14y=8}}} Multiply.



{{{-14y=8-8}}} Subtract {{{8}}} from both sides.



{{{-14y=0}}} Combine like terms on the right side.



{{{y=(0)/(-14)}}} Divide both sides by {{{-14}}} to isolate {{{y}}}.



{{{y=0}}} Reduce.



So the solutions are {{{x=1}}} and {{{y=0}}}.



Which form the ordered pair *[Tex \LARGE \left(1,0\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(1,0\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,-10,10,
grid(1),
graph(500,500,-9,11,-10,10,(4-4x)/(-7),(5-5x)/(2)),
circle(1,0,0.05),
circle(1,0,0.08),
circle(1,0,0.10)
)}}} Graph of {{{4x-7y=4}}} (red) and {{{5x+2y=5}}} (green)