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

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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) About Me  (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

2%2Ax-1%2Ay=12
1%2Ax%2B3%2Ay=-1

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

-1%2Ay=12-2%2AxSubtract 2%2Ax from both sides

y=%2812-2%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-12%2B2%2Ax Now we've fully isolated y

Since y equals -12%2B2%2Ax we can substitute the expression -12%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B3%2Ahighlight%28%28-12%2B2%2Ax%29%29=-1 Replace y with -12%2B2%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B3%2A%28-12%29%2B3%282%29x=-1 Distribute 3 to -12%2B2%2Ax

1%2Ax-36%2B6%2Ax=-1 Multiply



1%2Ax-36%2B6%2Ax=-1 Reduce any fractions

1%2Ax%2B6%2Ax=-1%2B36Add 36 to both sides


1%2Ax%2B6%2Ax=35 Combine the terms on the right side



7%2Ax=35 Now combine the terms on the left side.


cross%28%281%2F7%29%287%2F1%29%29x=%2835%2F1%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7%2F1 and isolate x

So when we multiply 35%2F1 and 1%2F7 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

1%285%29%2B3%2Ay=-1 Plug in x=5 into the 2nd equation

5%2B3%2Ay=-1 Multiply

3%2Ay=-1-5Subtract 5 from both sides

3%2Ay=-6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%28-6%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=-6%2F3 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=5 and y=-2

which can also look like

(5,-2)

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

2%2Ax-1%2Ay=12
1%2Ax%2B3%2Ay=-1

we get


graph of 2%2Ax-1%2Ay=12 (red) and 1%2Ax%2B3%2Ay=-1 (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 (5,-2). This verifies our answer.


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Check:

Plug in (5,-2) into the system of equations


Let x=5 and y=-2. Now plug those values into the equation 2%2Ax-1%2Ay=12

2%2A%285%29-1%2A%28-2%29=12 Plug in x=5 and y=-2


10%2B2=12 Multiply


12=12 Add


12=12 Reduce. Since this equation is true the solution works.


So the solution (5,-2) satisfies 2%2Ax-1%2Ay=12



Let x=5 and y=-2. Now plug those values into the equation 1%2Ax%2B3%2Ay=-1

1%2A%285%29%2B3%2A%28-2%29=-1 Plug in x=5 and y=-2


5-6=-1 Multiply


-1=-1 Add


-1=-1 Reduce. Since this equation is true the solution works.


So the solution (5,-2) satisfies 1%2Ax%2B3%2Ay=-1


Since the solution (5,-2) satisfies the system of equations


2%2Ax-1%2Ay=12
1%2Ax%2B3%2Ay=-1


this verifies our answer.





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