Select the correct answer.

Find the intersection point for the following linear functions.

[tex]\[
\begin{array}{l}
f(x) = 2x + 3 \\
g(x) = -4x - 27
\end{array}
\][/tex]

A. [tex]\((-5, -10)\)[/tex]
B. [tex]\((5, -7)\)[/tex]
C. [tex]\((5, 13)\)[/tex]
D. [tex]\((-5, -7)\)[/tex]



Answer :

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]