Sure, let's solve the problem step by step.
We are given two equations:
1. [tex]\(-2x + 4y = 15\)[/tex],
2. [tex]\(3x + y = 2\)[/tex].
The problem asks us to multiply the second equation by [tex]\(-4\)[/tex] and determine the coefficient that would appear in place of [tex]\(3x\)[/tex] when multiplied by [tex]\(-4\)[/tex] and placed in the corresponding position in the new equation.
Let's start by writing the second equation:
[tex]\[ 3x + y = 2 \][/tex]
Step-by-step multiplication by [tex]\(-4\)[/tex]:
1. Multiply the [tex]\(3x\)[/tex] term by [tex]\(-4\)[/tex]:
[tex]\[ -4 \cdot 3x = -12x \][/tex]
2. Multiply the [tex]\(y\)[/tex] term by [tex]\(-4\)[/tex]:
[tex]\[ -4 \cdot y = -4y \][/tex]
3. Multiply the constant term [tex]\(2\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[ -4 \cdot 2 = -8 \][/tex]
Thus, after multiplying the second equation [tex]\(3x + y = 2\)[/tex] by [tex]\(-4\)[/tex], the new equation will be:
[tex]\[ -12x - 4y = -8 \][/tex]
The value that belongs in the green box is the coefficient of the [tex]\(x\)[/tex] term in this new equation, which is [tex]\(\boxed{-12}\)[/tex].
