Given that m=(2,-7) and n =(5,1) solve the following vector equation @ xm+yn=(22,-3) ?

To solve the vector equation xm + yn = (22, -3), we need to find the values of x and y that satisfy the equation.

Given that m = (2, -7) and n = (5, 1), we can substitute these values into the equation:

x(2, -7) + y(5, 1) = (22, -3)

Expanding the equation, we get:

(2x, -7x) + (5y, y) = (22, -3)

Now, we can separate the components of the vectors:

2x + 5y = 22
-7x + y = -3

Now, we have a system of two equations with two variables. We can solve this system using any method, such as substitution or elimination.

From the first equation, we can solve for y:

y = (22 - 2x) / 5

Now, we can substitute this expression for y into the second equation:

-7x + (22 - 2x) / 5 = -3

Multiplying by 5 to get rid of the fraction:

-35x + 22 - 2x = -15

Combining like terms:

-37x = -37

Dividing by -37:

x = 1

Now, we can substitute x = 1 back into the first equation to solve for y:

2(1) + 5y = 22
2 + 5y = 22
5y = 20
y = 4

Therefore, the solution to the vector equation xm + yn = (22, -3) is x = 1 and y = 4.