To find the intersection point of the two linear functions [tex]\( f(x) = 2x + 3 \)[/tex] and [tex]\( g(x) = -4x - 27 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where these two functions are equal. This involves solving the system of equations derived from setting the two functions equal to each other.
1. Setting the equations equal to each other:
[tex]\[
2x + 3 = -4x - 27
\][/tex]
2. Combining like terms to solve for [tex]\( x \)[/tex]:
[tex]\[
2x + 4x = -27 - 3
\][/tex]
[tex]\[
6x = -30
\][/tex]
[tex]\[
x = \frac{-30}{6}
\][/tex]
[tex]\[
x = -5
\][/tex]
3. Substituting [tex]\( x = -5 \)[/tex] back into either function to find [tex]\( y \)[/tex]:
We'll use the function [tex]\( f(x) \)[/tex] to find [tex]\( y \)[/tex].
[tex]\[
f(x) = 2x + 3
\][/tex]
[tex]\[
f(-5) = 2(-5) + 3
\][/tex]
[tex]\[
f(-5) = -10 + 3
\][/tex]
[tex]\[
f(-5) = -7
\][/tex]
So the [tex]\( y \)[/tex]-coordinate is [tex]\(-7\)[/tex].
4. Conclusion:
The intersection point of the two functions is [tex]\((-5, -7)\)[/tex].
Upon reviewing the given options, the correct answer is:
[tex]\[ \boxed{(-5, -7)} \][/tex]
