SOLUTION: Express the vector v= [-17] as a linear combination of x = [-2] and y = [-5] [ 50 ]

Question 629335:
Express the vector v= [-17] as a linear combination of x = [-2] and y = [-5]
[ 50 ] [ 5 ] [ -6]
v=__x + __ y.
How could I express this?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
I'm assuming that , and .

The problem is to find what number times x plus what number times y adds up to v. Let's call these two numbers a and b. So we're looking for what a and b make
v = a*x + b*y
true.

Substituting the vectors/column matrices in for v, x and y out equation is:

Performing the scalar multiplication on the right side we get:

Adding the vectors/column matrices on the right we get:

By the definition of matrix/vector equality, these are equal only if each element of one is equal to the corresponding element of the other. So
-17 = -2a + (-5b) and
50 = 5a + (-6b)

To find the correct values for a and b we solve this system. You have probably learned several ways to solve a system of linear equations like this: Substitution, Linear Combination/Elimination, several matrix methods, determinants/Cramer's Rule, etc. I'm going to use Linear Combination (but any of them will work and give the right answers).
Multiplying the first equation by 5 and the second equation by 2 we get:
-85 = -10a + (-25b)
100 = 10a + (-12b)
Adding the two equations together we get:
15 = -37b
Dividing both sides by -37 we get:
-15/37 = b
Now we use this value and one of the earlier equations to find a:

Simplifying we get:

Subtracting 75/37 to each side:

Dividing by -2:

RELATED QUESTIONS
easy calc - Express vector w as...? i know this is supposed to be a really... (answered by jim_thompson5910)

Consider the following two vectors in R3 : u=(1,2,3) and v=(2,3,1). Show whether it is... (answered by Fombitz)

If v = <-2, 5> and w = <3, 4>, then v + 2w = ? Write your answer as a vector in the... (answered by ikleyn)

Express v=(1,-2,5)∈ R³ as a linear combination of vectors u₂=(1,1,1),... (answered by robertb)

Calculate gcd(x,y) and express it as an integral linear combination of x and y. a)... (answered by venugopalramana)

The vector v=[4 5] can be written as a linear combination of the vectors v1=[1 3]... (answered by Edwin McCravy)

Let two points be A(2, 5) and B(-3, 6). Express the vector determined by line segment... (answered by Edwin McCravy)

17. In each part, sketch the vector u + v+ w, and express it in component form.... (answered by stanbon,TipoTropical)

Given V=R^2 with "non-standard" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real (answered by venugopalramana)