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] that satisfy both equations simultaneously. The intersection point occurs when [tex]\( f(x) = g(x) \)[/tex].
1. Set the two equations equal to each other:
[tex]\[
2x + 3 = -4x - 27
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Combine like terms by adding [tex]\( 4x \)[/tex] to both sides:
[tex]\[
2x + 4x + 3 = -27
\][/tex]
This simplifies to:
[tex]\[
6x + 3 = -27
\][/tex]
Next, subtract 3 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
6x = -27 - 3
\][/tex]
[tex]\[
6x = -30
\][/tex]
Now, divide both sides by 6:
[tex]\[
x = \frac{-30}{6}
\][/tex]
[tex]\[
x = -5
\][/tex]
3. Substitute [tex]\( x = -5 \)[/tex] back into either of the original equations to find [tex]\( y \)[/tex]. We'll use [tex]\( f(x) = 2x + 3 \)[/tex]:
[tex]\[
f(-5) = 2(-5) + 3
\][/tex]
[tex]\[
f(-5) = -10 + 3
\][/tex]
[tex]\[
f(-5) = -7
\][/tex]
4. Thus, the intersection point [tex]\((x, y)\)[/tex] is:
[tex]\[
(-5, -7)
\][/tex]
So, the correct answer is:
D. [tex]\((-5, -7)\)[/tex]
