To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given that the vectors [tex]\( \vec{a} = \binom{4}{-5} \)[/tex] and [tex]\( \vec{b} = \binom{2x}{3+y} \)[/tex] are equal, we equate their corresponding components. This means that each component of [tex]\(\vec{a}\)[/tex] must equal the corresponding component of [tex]\(\vec{b}\)[/tex].
Given:
[tex]\[ \vec{a} = \binom{4}{-5} \][/tex]
[tex]\[ \vec{b} = \binom{2x}{3+y} \][/tex]
Since the vectors are equal, their corresponding components must be equal:
[tex]\[ \binom{4}{-5} = \binom{2x}{3+y} \][/tex]
This results in the following systems of equations:
1. [tex]\( 4 = 2x \)[/tex]
2. [tex]\( -5 = 3 + y \)[/tex]
To solve for [tex]\( x \)[/tex]:
[tex]\[ 4 = 2x \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{4}{2} \][/tex]
[tex]\[ x = 2 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ -5 = 3 + y \][/tex]
Subtract 3 from both sides:
[tex]\[ y = -5 - 3 \][/tex]
[tex]\[ y = -8 \][/tex]
So the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that make the vectors [tex]\(\vec{a} \)[/tex] and [tex]\(\vec{b} \)[/tex] equal are:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = -8 \][/tex]
