Question 171367

Start with the given system of equations:

{{{system(x+5y=17,-4x-3y=24)}}}



{{{4(x+5y)=4(17)}}} Multiply the both sides of the first equation by 4. This will make the "x" coefficients equal but opposite (which will mean that they will cancel).



{{{4x+20y=68}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4x+20y=68,-4x-3y=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:



{{{(4x+20y)+(-4x-3y)=(68)+(24)}}}



{{{(4x+-4x)+(20y+-3y)=68+24}}} Group like terms.



{{{0x+17y=92}}} Combine like terms. Notice how the x terms cancel out.



{{{17y=92}}} Simplify.



{{{y=(92)/(17)}}} Divide both sides by {{{17}}} to isolate {{{y}}}.



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{{{4x+20y=68}}} Now go back to the first equation.



{{{4x+20(92/17)=68}}} Plug in {{{y=92/17}}}.



{{{4x+1840/17=68}}} Multiply.



{{{17(4x)+cross(17)(1840/cross(17))=17(68)}}} Multiply EVERY term by the LCD {{{17}}} to clear any fractions.



{{{68x+1840=1156}}} Distribute and multiply.



{{{68x=1156-1840}}} Subtract {{{1840}}} from both sides.



{{{68x=-684}}} Combine like terms on the right side.



{{{x=(-684)/(68)}}} Divide both sides by {{{68}}} to isolate {{{x}}}.



{{{x=-171/17}}} Reduce.



So the solutions are {{{x=-171/17}}} and {{{y=92/17}}}.



Which form the ordered pair *[Tex \LARGE \left(-\frac{171}{17},\frac{92}{17}\right)].



This means that the system is consistent and independent.