To find the value of [tex]\( k \)[/tex] such that the system of equations formed by the functions [tex]\( g \)[/tex] and [tex]\( h \)[/tex] has infinitely many solutions, follow these steps:
1. Define the given functions:
[tex]\[
f(x) = \frac{4}{3} x - 3
\][/tex]
[tex]\[
h(x) = ax + 2
\][/tex]
2. Define the function [tex]\( g \)[/tex], which is [tex]\( f(x) \)[/tex] moved up by [tex]\( k \)[/tex] units:
[tex]\[
g(x) = f(x) + k = \frac{4}{3} x - 3 + k
\][/tex]
3. For the system constituted by [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have infinitely many solutions, [tex]\( g \)[/tex] must be identical to [tex]\( h \)[/tex]. This means that both the slopes and the y-intercepts of [tex]\( g \)[/tex] and [tex]\( h \)[/tex] must be equal.
4. Compare the slopes:
The slope of [tex]\( f(x) \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
The slope of [tex]\( g(x) \)[/tex] is also [tex]\( \frac{4}{3} \)[/tex] because it is simply [tex]\( f(x) \)[/tex] shifted vertically.
The slope of [tex]\( h(x) \)[/tex] is [tex]\( a \)[/tex].
For [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have the same slope:
[tex]\[
\frac{4}{3} = a \implies a = \frac{4}{3}
\][/tex]
5. Compare the y-intercepts:
The y-intercept of [tex]\( f(x) \)[/tex] is [tex]\( -3 \)[/tex].
The y-intercept of [tex]\( g(x) \)[/tex], after moving up [tex]\( k \)[/tex] units, is:
[tex]\[
-3 + k
\][/tex]
The y-intercept of [tex]\( h(x) \)[/tex] is [tex]\( 2 \)[/tex].
For [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have the same y-intercept:
[tex]\[
-3 + k = 2 \implies k = 2 + 3 \implies k = 5
\][/tex]
So, the value of [tex]\( k \)[/tex] is:
[tex]\[
k = 5
\][/tex]
