Question 629335
I'm assuming that {{{v = (matrix(2, 1, -17, 50))}}}, {{{x = (matrix(2, 1, -2, 5))}}} and {{{y = (matrix(2, 1, -5, -6))}}}.<br>
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.<br>
Substituting the vectors/column matrices in for v, x and y out equation is:
{{{(matrix(2, 1, -17, 50)) = a*(matrix(2, 1, -2, 5)) + b*(matrix(2, 1, -5, -6))}}}
Performing the scalar multiplication on the right side we get:
{{{(matrix(2, 1, -17, 50)) = (matrix(2, 1, -2a, 5a)) + (matrix(2, 1, -5b, -6b))}}}
Adding the vectors/column matrices on the right we get:
{{{(matrix(2, 1, -17, 50)) = (matrix(2, 1, -2a + (-5b), 5a + (-6b)))}}}
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)<br>
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:
{{{-17 = -2a + (-5)(-15/37)}}}
Simplifying we get:
{{{-17 = -2a + 75/37}}}
Subtracting 75/37 to each side:
{{{-17 - 75/37 = -2a }}}
{{{-629/37 - 75/37 = -2a}}}
{{{-704/37 = -2a}}}
Dividing by -2:
{{{352/37 = a}}}<br>
So
{{{v = (352/37)x + (-15/37)y}}}