Find the values of x and y, if [tex]x \begin{pmatrix} -1 \\ 3 \end{pmatrix} + y \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ -8 \end{pmatrix}[/tex]
Answer (1)
To solve the system of equations given by the matrix equation:
x ( − 1 3 ) + y ( 2 1 ) = ( − 1 − 8 )
we will equate the corresponding components of the vectors on both sides of the equation to create a system of linear equations.
From the first component:
− x + 2 y = − 1
From the second component:
3 x + y = − 8
Now, let's solve this system of equations.
First, let's express one variable in terms of the other using the second equation:
y = − 8 − 3 x
Substitute y = − 8 − 3 x into the first equation:
− x + 2 ( − 8 − 3 x ) = − 1
Simplify:
− x − 16 − 6 x = − 1
Combine like terms:
− 7 x − 16 = − 1
Add 16 to both sides:
− 7 x = 15
Divide both sides by -7:
x = − 7 15
Now, substitute x = − 7 15 back into y = − 8 − 3 x to find y :
y = − 8 − 3 ( − 7 15 )
Simplify:
y = − 8 + 7 45
Convert − 8 to a fraction with a denominator of 7:
y = 7 − 56 + 7 45
Combine the fractions:
y = 7 − 56 + 45 = 7 − 11
Therefore, the values of x and y are:
x = − 7 15
y = − 7 11
