SOLUTION: easy calc - Express vector w as...? i know this is supposed to be a really easy question but i can't get to the right answer!! i know you're supposed to find a/b by eli

Question 99659: easy calc - Express vector w as...?
i know this is supposed to be a really easy question but i can't get to the right answer!! i know you're supposed to find a/b by elimination then sub but i keep getting the wrong answer...
heres the question:
Given the vectors u = [2,1] and v= [-1,3]:
a) Express the vector w = [12, -1] as a linear combination of vector u and v.
ANSWER: 5u - 2v
can someone explain / show work of how to do this please!!!
PS. all the letters have the little vector sign on them, just dunno how to do it on the computer lol.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Express the linear combination as a system of equations

2x-y=12
-x+3y=-1

Now solve the system using substitution:

Solved by pluggable solver: Solving a linear system of equations by subsitution

Lets start with the given system of linear equations

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

Solve for y for the first equation

Subtract from both sides

Divide both sides by -1.

Which breaks down and reduces to

Now we've fully isolated y

Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x.

Replace y with . Since this eliminates y, we can now solve for x.

Distribute 3 to

Multiply

Reduce any fractions

Add to both sides

Combine the terms on the right side

Now combine the terms on the left side.

Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get

<---------------------------------One answer

Now that we know that , lets substitute that in for x to solve for y

Plug in into the 2nd equation

Multiply

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 3 on the left side.

Multiply the terms on the right side

Reduce

So this is the other answer

<---------------------------------Other answer

So our solution is

and

which can also look like

(,)

Notice if we graph the equations (if you need help with graphing, check out this solver)

we get

graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.

and we can see that the two equations intersect at (,). This verifies our answer.

-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Since the solution (,) satisfies the system of equations

this verifies our answer.

So x=5, y=-2 translates to 5u-2v
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