Question 203481
# 1


{{{y=-3x+19}}} Start with the first equation.



{{{2x-1=-3x+19}}} Plug in {{{y=2x-1}}}



{{{2x=-3x+19+1}}} Add {{{1}}} to both sides.



{{{2x+3x=19+1}}} Add {{{3x}}} to both sides.



{{{5x=19+1}}} Combine like terms on the left side.



{{{5x=20}}} Combine like terms on the right side.



{{{x=(20)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



Now that we know the value of 'x', we can solve for 'y'



{{{y=-3x+19}}} Go back to the first equation.



{{{y=-3(4)+19}}} Plug in {{{x=4}}}



{{{y=-12+19}}} Multiply



{{{y=7}}} Combine like terms.



So the solutions are {{{x=4}}} and {{{y=7}}}.



Which form the ordered pair *[Tex \LARGE \left(4,7\right)].



This means that the system is consistent and independent.




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# 2


Start with the given system of equations:

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



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



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



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



{{{0x+4y=3}}} Combine like terms. Notice how the x terms cancel out.



{{{4y=3}}} Simplify.



{{{y=3/4}}} Divide both sides by {{{4}}} to isolate {{{y}}}.



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



{{{x+3(3/4)=2}}} Plug in {{{y=3/4}}}.



{{{x+9/4=2}}} Multiply.



{{{4(x+9/cross(4))=4(2)}}} Multiply both sides by the LCD {{{4}}} to clear any fractions.



{{{4x+9=8}}} Distribute and multiply.



{{{4x=8-9}}} Subtract 9 from both sides.



{{{4x=-1}}} Combine like terms.



{{{x=-1/4}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



So the solutions are {{{x=-1/4}}} and {{{y=3/4}}}.



Which form the ordered pair *[Tex \LARGE \left(-\frac{1}{4},\frac{3}{4}\right)].



This means that the system is consistent and independent.