To determine the correct result of substituting for [tex]\( y \)[/tex] in the bottom equation, let's start by analyzing the two given equations:
1. [tex]\( y = x + 3 \)[/tex]
2. [tex]\( y = x^2 + 2x - 4 \)[/tex]
We need to substitute [tex]\( y \)[/tex] from the first equation into the second equation. The substitution process works as follows:
1. From the first equation, we know that [tex]\( y = x + 3 \)[/tex].
2. Substitute [tex]\( y = x + 3 \)[/tex] into the second equation:
[tex]\[
y = x^2 + 2x - 4
\][/tex]
This becomes:
[tex]\[
x + 3 = x^2 + 2x - 4
\][/tex]
Let's check the options given:
- Option A: [tex]\( y = (x + 3)^2 + 2(x + 3) - 4 \)[/tex]
This option introduces an incorrect manipulation and unnecessary squaring. It is not the result of simple substitution.
- Option B: [tex]\( x + 3 = x^2 \)[/tex]
This option simplifies the original substitution and omits critical terms, thus it is incorrect.
- Option C: [tex]\( y = x^2 + 2x - 4 - (x + 3) \)[/tex]
This option does not properly substitute for [tex]\( y \)[/tex]. Instead, it introduces a subtraction that isn’t present in the original substitution.
- Option D: [tex]\( x + 3 = x^2 + 2x - 4 \)[/tex]
This correctly shows the result of substituting [tex]\( y \)[/tex] from the first equation into the second equation.
The correct answer is thus:
D. [tex]\( x + 3 = x^2 + 2x - 4 \)[/tex]
