Question 183069
Since the "package contains the same amount of Hazelnut coffee as French roast coffee", this means that {{{y=z}}} is your third equation. Since you have 3 equations in 3 unknowns, you can find a unique solution (if there is one) to the system.





Here's the Updated Solution:



{{{x + y + z = 10}}} Start with the first equation.



{{{x + z + z = 10}}} Plug in {{{y=z}}}. In other words, replace each "y" with "z"



{{{x + 2z = 10}}} Combine like terms. Let's call this equation 4.



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{{{2x + 2.50y + 3z = 26}}} Move onto the second equation



{{{20x + 25y + 30z = 260}}} Multiply EVERY term by 10 to make every number a whole number.



{{{20x + 25z + 30z = 260}}} Plug in {{{y=z}}}



{{{20x + 55z = 260}}} Combine like terms. Let's call this equation 5.



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So we have the system of equations 4 and 5:


{{{system(x+2z=10,20x+55z=260)}}}



Let's solve this smaller system by substitution



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for z.




So let's isolate z in the first equation



{{{x+2z=10}}} Start with the first equation



{{{2z=10-x}}}  Subtract {{{x}}} from both sides



{{{2z=-x+10}}} Rearrange the equation



{{{z=(-x+10)/(2)}}} Divide both sides by {{{2}}}



{{{z=((-1)/(2))x+(10)/(2)}}} Break up the fraction



{{{z=(-1/2)x+5}}} Reduce




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Since {{{z=(-1/2)x+5}}}, we can now replace each {{{z}}} in the second equation with {{{(-1/2)x+5}}} to solve for {{{x}}}




{{{20x+55highlight(((-1/2)x+5))=260}}} Plug in {{{z=(-1/2)x+5}}} into the second equation. In other words, replace each {{{z}}} with {{{(-1/2)x+5}}}. Notice we've eliminated the {{{z}}} variables. So we now have a simple equation with one unknown.




{{{20x+(55)(-1/2)x+(55)(5)=260}}} Distribute {{{55}}} to {{{(-1/2)x+5}}}



{{{20x-(55/2)x+275=260}}} Multiply



{{{(2)(20x-(55/2)x+275)=(2)(260)}}} Multiply both sides by the LCM of 2. This will eliminate the fractions.



{{{40x-55x+550=520}}} Distribute and multiply the LCM to each side




{{{-15x+550=520}}} Combine like terms on the left side



{{{-15x=520-550}}}Subtract 550 from both sides



{{{-15x=-30}}} Combine like terms on the right side



{{{x=(-30)/(-15)}}} Divide both sides by -15 to isolate x




{{{x=2}}} Divide




Since we know that {{{x=2}}} we can plug it into the equation {{{z=(-1/2)x+5}}} (remember we previously solved for {{{z}}} in the first equation).




{{{z=(-1/2)x+5}}} Start with the equation where {{{z}}} was previously isolated.



{{{z=(-1/2)(2)+5}}} Plug in {{{x=2}}}



{{{z=-2/2+5}}} Multiply



{{{z=4}}} Combine like terms and reduce.  





Now because we know that {{{y=z}}} and {{{z=4}}}, this tells us that {{{y=4}}} also





========================= Answer =============================



So the solutions are


{{{x=2}}}, {{{y=4}}} and {{{z=4}}}



which form the ordered triple (2,4,4)





This means that there are 2 lbs of Vanilla flavored coffee, 4 lbs of Hazelnut flavored coffee, and 4 lbs of French Roast flavored coffee