To determine the value of [tex]\( b \)[/tex] in the linear equation [tex]\( a x + b y = 12 \)[/tex] given that the line passes through the points [tex]\((0,3)\)[/tex] and [tex]\((2,-4)\)[/tex], let's proceed step-by-step:
1. Substitute the points into the equation:
For the point [tex]\((0, 3)\)[/tex]:
[tex]\[
a \cdot 0 + b \cdot 3 = 12
\][/tex]
Simplifying this equation:
[tex]\[
3b = 12
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = \frac{12}{3} = 4
\][/tex]
2. Verify the value using the second point [tex]\((2, -4)\)[/tex]:
Substitute [tex]\( b = 4 \)[/tex] into the equation for the second point [tex]\((2, -4)\)[/tex]:
[tex]\[
a \cdot 2 + 4 \cdot (-4) = 12
\][/tex]
Simplifying this equation:
[tex]\[
2a - 16 = 12
\][/tex]
Rearrange terms to solve for [tex]\( a \)[/tex]:
[tex]\[
2a = 12 + 16
\][/tex]
[tex]\[
2a = 28
\][/tex]
[tex]\[
a = \frac{28}{2} = 14
\][/tex]
Now, we have confirmed that the value of [tex]\( a \)[/tex] is consistent with [tex]\( b = 4 \)[/tex].
Thus, the value of [tex]\( b \)[/tex] is
[tex]\[
\boxed{4}
\][/tex]
