Question 201817


Start with the given system of equations:

{{{system(4x-7y=-5,5x+3y=29)}}}



{{{3(4x-7y)=3(-5)}}} Multiply the both sides of the first equation by 3.



{{{12x-21y=-15}}} Distribute and multiply.



{{{7(5x+3y)=7(29)}}} Multiply the both sides of the second equation by 7.



{{{35x+21y=203}}} Distribute and multiply.



So we have the new system of equations:

{{{system(12x-21y=-15,35x+21y=203)}}}



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



{{{(12x-21y)+(35x+21y)=(-15)+(203)}}}



{{{(12x+35x)+(-21y+21y)=-15+203}}} Group like terms.



{{{47x+0y=188}}} Combine like terms.



{{{47x=188}}} Simplify.



{{{x=(188)/(47)}}} Divide both sides by {{{47}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



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{{{12x-21y=-15}}} Now go back to the first equation.



{{{12(4)-21y=-15}}} Plug in {{{x=4}}}.



{{{48-21y=-15}}} Multiply.



{{{-21y=-15-48}}} Subtract {{{48}}} from both sides.



{{{-21y=-63}}} Combine like terms on the right side.



{{{y=(-63)/(-21)}}} Divide both sides by {{{-21}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



So the solutions are {{{x=4}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(4,3\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(4,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-6,14,-7,13,
grid(1),
graph(500,500,-6,14,-7,13,(-5-4x)/(-7),(29-5x)/(3)),
circle(4,3,0.05),
circle(4,3,0.08),
circle(4,3,0.10)
)}}} Graph of {{{4x-7y=-5}}} (red) and {{{5x+3y=29}}} (green)