Question 241414


Start with the given system of equations:

{{{system(7x+24y=24,3x-4y=-4)}}}



{{{6(3x-4y)=6(-4)}}} Multiply the both sides of the second equation by 6.



{{{18x-24y=-24}}} Distribute and multiply.



So we have the new system of equations:

{{{system(7x+24y=24,18x-24y=-24)}}}



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



{{{(7x+24y)+(18x-24y)=(24)+(-24)}}}



{{{(7x+18x)+(24y+-24y)=24+-24}}} Group like terms.



{{{25x+0y=0}}} Combine like terms.



{{{25x=0}}} Simplify.



{{{x=(0)/(25)}}} Divide both sides by {{{25}}} to isolate {{{x}}}.



{{{x=0}}} Reduce.



------------------------------------------------------------------



{{{7x+24y=24}}} Now go back to the first equation.



{{{7(0)+24y=24}}} Plug in {{{x=0}}}.



{{{0+24y=24}}} Multiply.



{{{24y=24-0}}} Subtract {{{0}}} from both sides.



{{{24y=24}}} Combine like terms on the right side.



{{{y=(24)/(24)}}} Divide both sides by {{{24}}} to isolate {{{y}}}.



{{{y=1}}} Reduce.



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



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



{{{drawing(500,500,-10,10,-9,11,
grid(1),
graph(500,500,-10,10,-9,11,(24-7x)/(24),(-4-3x)/(-4)),
circle(0,1,0.05),
circle(0,1,0.08),
circle(0,1,0.10)
)}}} Graph of {{{7x+24y=24}}} (red) and {{{3x-4y=-4}}} (green)